At what point do I overrule my parent's wishes?

There are really two issues here: when should you make the decision on their behalf, and when can you make that decision.

Since the doctor is worried about him living alone, I take it there is no spouse involved. Is that correct?

There are 3 options:
1. Continue living at home, with round-the-clock care. This option tends to be the most expensive.
2. Move in with a family member. This is an option he has specifically rejected in the past.
3. Move into a care facility with the appropriate level of care and supervision. Apparently this is the option your parent would prefer.

Surprisingly, living with family members is not always the best option. Sometimes it is, of course, but not as often as you might suppose.

If he were in the care center you have in mind, could a family member take him to the swim classes? Go on outings with him often? Consider that he probably wouldn't be the only high-functioning person there, and those residents tend to find each other.

When should the decision be made? Now. He is no longer safe to live alone.
What can you do to make the decision for him? Persuasion only. He is high-functioning enough to make his own decision.

Often, if you wait too long, the window of opportunity is lost forever--
Moving slowly and methodically as you have has given you the chance to check our several places. Now take dad and mom to visit.
My old 'Client" when I worked in elder care, was ready to go---but she didn't know it! It took slow, gentle persuasion and MANY trips to the new "facility" to acclimate, before long, she accepted the family request that she move--and she thought it was her own decision!
As far as outings, ALF patients can come and go. My client routinely went to parties, luncheons, family activities...no big deal. Had she wanted to go swimming I am sure they would have made that happen.
I will add---her place was the nicest, most posh ALF I've ever been in. The food was amazing, the staff, top notch and her little apartment, just delightful. Her family spared NO expense, and often told me it was worth every penny.

Within a year of being moved, her dementia worsened, she would not have done so well in that condition, being moved from home. Eventually she did wind up in a lockdown type area of the same ALF. Still nice, but dedicated to keeping the patient safe--and from wandering.

I'd strike while the iron is hot. We waited to long to "push" Mother along and now she doesn't have the ability to make the decision and as kids, we can't do it for her, as we cannot all agree, and never will.

Good luck.

Are you sure they would be miserable? Does this facility have levels of care? In other words, would the resident be required to be in a locked facility or are there sections with more freedom? If they can still participate in the community, they might feel happier and more free in the facility rather than in your home.

From many a caregivers point of view - including mine -  most would consider you lucky.

Lucky that you have a parent that does NOT want to be taking in and taken care of by their family. In particular- one of their children. Lucky that you have a parent that has the means to finance their choice of living arrangements and lifestyle. And lucky that your parent has remained active both physically and mentally- as well as being active and perhaps practical in their own decisions.

While you look at this choice to move into a facility as a potentially over restrictive one - could it be that your parent views it as a step in maintaining some independence - not having to be "taken care of" by their child. Perhaps they are not willing to let the often occurrence of role reversal with the child taking care of the parent take place any sooner than it needs to when dementia is involved. 

As well, could dignity and respect factor into this choice. Of course you would treat your parent with both dignity and respect - but perhaps your parent is not willing or wanting you and your family to witness their deminishing skills and abilities. And deminish they will.

To your parent this move may be in their eyes the best shot and keeping your relationship one of family - of parent and child verse caregiver and "patient".

There are multiple smaller issues as well. Maybe they'd like to remain in their own generational peer group. Maybe they believe there will be more activities available to them for keeping busy. Maybe they would like more personal space verse having family around all day, everyday.

There are a whole lot of maybes. But if this is your parents choice - personally, I think you ought to give them their earned right to make it. After all, if it doesn't work out you could always bring them to your home.

All good answers IMHO. I would specify an event that he alone controls as to when he goes to the nursing home.
For my dad who was 100% against ANY nursing home, the tipping point was when he had an incident that involved any harm or injury. Lacking that I, my brothers and mother, felt it was still his decision. It didn't matter that dementia was taking hold, that he could no longer decipher the microwave instructions on the box, that we had to lay his medicine out and he would pick and choose which to take, we felt the time was just not right. He still had reasoning to control his destiny. But when he lost his way coming home from the $ store, [6 blocks from home]- ran the car over a curb 18 miles from home, it was time.
He still periodically asks why he is there, and we tell him. He promptly forgets then later, maybe that visit or the next one, he asks all over again. He is now getting the care and safe living quarters he needs.
I hope someone has been appointed as POA and he has an up to date will made. Too late now to put any real property into a trust.

One thing you may want to consider if someone has Alzheimer's is taking away the keys. We have a local who was caught going the wrong way on I 71 and her license was immediately and permanently revoked when they found out she was demented. She lives not far from me right now but her family must make sure she never drives again because she could've killed someone that day or maybe even herself. This person may also still be high functioning but not capable of driving. A simple mistake behind the wheel can spell disaster. Another area you'll want to watch over is how your elder is spending their money or if someone is taking advantage of them. They may be high functioning, but that doesn't necessarily mean someone won't use that to their advantage

In my view, you over rule when it becomes a matter of your parent's safety. When parent's wishes go against parent's safety, safety has to come first.
My mother has a personality disorder and developed vascular dementia the past few years. She became paranoid about staff at the very nice ALF where she lived, and was very well cared for, and found another ALF she wanted to live in. I (POA medical and financial) looked into it and found that services were less than she had where she was and it would not be a suitable place for her. For example, there was no nurse on call on her floor, just a security guard. This was not enough for a 100 year old! She was quite upset at me when I said I could not stop her from moving, but I would not support or help that move, as I felt it was not in her best interests. In time and after many difficulties, she was hospitalized and diagnosed with the vascular dementia, treated and placed in a facility very suitable to her needs.

I have a very good friend who was in this position. Although skeptical at first, she was glad she visited and made the decision to move into a care facility, since she did not want to live with relatives or continue to live in her home after her husband died. She greatly enjoyed living in a senior care facility, where you can live independently, and when needed, receive care at every level. The grounds are beautiful, there are educational and recreational activities, a lovely dining area, etc. She made the choice to move there, and to be as involved or not involved as she wished. It was safe and provided the care she needed. She moved in when she was very independent and still drove. As she needed more care, she moved to another facility of this kind, closer to her family.
Since you are asking about making this decision, take cues from the doctor and your parent's wishes. Take him/her to the facility and explore that option, discuss the option of having care in his/her home, as well as coming to live with you. Give your parent time to think about it. It may help to try in home care first, especially if he/she is safe at night, before moving on to another option. Each person is different, but having doctor input and considering your parent's needs and wishes as well as your peace of mind, is important.
Best wishes for making this decision.

I would honor my parent's wishes in this case. Caregiving is extremely hard and they made the right decision way back not to burden a relative.

I was facing the same dilemma with my two friends for whom I am their POA. The husband was adamant that they could still live independently. Both had had their licenses revoked, so I took them shopping once a week for groceries and to their appointments and occasional religious activities. When the wife became incontinent and started to wander, due to her frontal temporal dementia, the part time people I had hired to help with her personal care told me she needed 24 hour care, otherwise I could be charged with neglect. I finally was able to persuade the husband to go to an assisted living facility with a memory care floor with the choice of a two bedroom, one bedroom or efficiency apartment. They had to go as a couple, so I picked out a one bedroom apartment for them and the day of the move, arranged to have the furniture they would use brought from their town home to the new apartment and set up just like it had been at home. The husband liked it from the beginning and never talked about going back to their town home. The care there was very good, along with the advice they gave me about how the dementia was progressing. I needed that since I have never done this before. The AL portion has all sorts of daily activities for the residents. The memory care level has their own activities, which the husband does not care to partake of. He is happy watching TV and reading the daily paper and socializing just at meals. It sounds like you have done some good research and if the facility is anything like the one I found, he will be well cared for. They like to do things to stimulate the mind and there are occasional outings, too, for the AL residents. The husband very much has his identity intact and likes living there and his new meal friends. His wife passed on in 2015 and he misses her, but accepts that she had to go. He feels very fortunate to be in this facility and the good fortune he had in meeting and marrying such a person as his wife, and to have me as a friend watching over everything. I am grateful to have found this one place that does so many things so very well. They told me he will never have to leave and they will accept public financing should he run out of money (after 18 months of regular payments) and they provide care to the very end with hospice when necessary. Since living alone is no longer an option. a good AL-MC facility can be excellent choice.