How do I say no to another pet?

Who are the people who predicted that your father would want another dog? Are any of them local to him and prepared to help on a daily basis?

Companion animals give their owners a reason to get up in the morning, manage a schedule, walk around in the fresh air, and think all day long about something other than their own situation. Their value in terms of quality of life is immeasurable. But it is also reasonable for you to decide that you can't manage the additional workload - so, can anyone else help with this?

There are dog dolls that look pretty realistic that are made for people with dementia. He can pet it and love on it.
there are robot ones and just stuffed ones.
https://www.alzstore.com/doll-pet-therapy-dementia-s/1516.htm

do you know anyone with a dog? There is an elderly gentleman down the road from my parent who always had dogs. Now that he is older he doesn't want to risk getting another pup and potentially leave them with no one. So he walks the dog of someone in the neighbourhood every day. The dog also goes for visits. It is so sweet to see them together.. best buds :) Plus everyone benefits from the arrangement
Maybe a dog 'timeshare' is possible.

Wmy does everyone try to bend over backwards to appease the elder? The poster is clearly burnt out and does not want to take on the responsibility of another pet. Dog or cat. Not even to foster.

I completely understand. Just ignore his asking for a pet or divert it to another day and maybe he will forget about it. Let's support the OP and not try and give her options and alternatives she is not up for.

Animals should be in homes where they are wanted and cared for. Sorry the dad wants another dog but we can't always get what we want. I like the idea of a stuffed animal but it sounds like he is not in a state right now where this would appease him.

Many shelters are now looking for Foster Homes.
You do not have the medical cost you would if it were your dog. (although you might have to take it to one of the "approved" vets if it is getting current care.
I would also, if you are going to do this, get one that is short haired and does not require routine visits to a groomer.
Some dogs can be littler box trained so that might be a possibility. Although they do need to be taken out for routine exercise.
Would your dad accept a cat as a companion? Far less to do with and for a cat.
Is dad living with you? If so your house , your rules so if you do not want another dog you have to say so.
If dad is living alone...(and he should not be with dementia) but if he has caregivers much of the tending to the dog can be part of their job description.
If he is living alone, if there is a possibility that he will have to make a move to Memory Care many facilities will not allow a pet unless the resident can care for it (or pay to have someone care for it) so not getting one now would make it easier if he has to transition to MC

I would not allow another dog. Tell him you have been doing all the care and you can no longer do it.

I do like the fostering idea. The problem may be though, fostered dogs get adopted. This means he will again lose a dog.

I know this sounds stupid - but have you considered a stuffed animal? A very nice, realistic stuffed dog? My brother worked for a while in a SNF with a memory care unit and he said that there were a number of residents there with baby dolls and stuffed animals - they considered them their children or their pets, emotional support animals. They took them everywhere with them and took care of them. Sometimes it is about being able to hold something. That may not work for everyone, but it may be worth a shot.

Are there any children in the neighborhood who would be willing to help with care of pet? Especially kids who cannot have pets in their own home.

If you are providing direct for a person with dementia don’t do it. You already have enough to do and stated that you aren’t up for more. I would never consider getting a dog or cat for an elder.

I would ask him about fostering.

This means that the rescue will handle all the vet visits and he could get one that needs rehab and isn't very active.

I understand how challenging it is for you but, him living alone means he is lacking companionship and a pet fills that void. You can't be everything to him, no one person can do that for another.

A younger dog would not have the same issues that an older one has.

Can dad afford a housekeeper? Make that part of the deal to introduce another dog, then find one that is willing to clean after a dog. Can he afford a dog walker?

My dads facility let him have his dog and she was a huge blessing to everyone in the house, maybe make finding a board and care home that accepts dogs an option.

I known that there comes a time when someone can no longer handle a pet on their own but, there are other options besides yes or no.

Please don't get another dog or cat. Your dad's dementia is only going to get worse and it wouldn't be fair to the animal as he gets worse.

I guess you can start some therapeutic lying here. Come up with something kind and vague. Like "I know it would be really nice to have a dog. I miss our old dog too. I'll keep looking but there just aren't any available right now." Could he be happy with an animal that is easier to care for? Like a cat? They are much more independent and don't need to be walked. They may curl up on his lap for hours and give him love and companionship. That would be a lot easier for you.