How do you know it's time to move parents from independent living to a nursing home?

It's not an easy decision to make. My mom went into the nursing home to rehab for other ailments when she was diagnosed with dementia. That diagnosis explained some of the things that had been happening at home lately. We feared for her safety at home (she lives with me and my wife, but we work and can not be home 24/7) as well as our own safety. She once dropped an iron on the floor that was on and hot, she did not realize it was on. What made the decision a little easier was that to have 24/7 professional help was the amount it would cost us. $16.00 an hour added up to appx. $140,000 a year. I have had guilty feeling over this, my stomach has been in knots and my head feels like it is spinning, however I listened to what the doctor, nurses, case worker, our elder law attorney and our funeral director advised and they all said that for moms health and safety putting her in a nursing home was best. It's not easy. Good luck in whatever you decide.

Ask their doctors to advise you about this.

I'm assuming you're talking about an actual independent living senior community (rather than a 55+ development). If so, I'd sit down with the director. Every community is different in its flexibility about independent residents needing care. But, if they have assisted living at the same site, they may offer home care for their residents who start to need help with ADLs (activities of daily living). Some allow you to contract your own, others require that you use their staff. Either way, it's going to be more pleasant, especially for your dad, to continue to live among his IL neighbors (not to mention less expensive!). If they just need a couple of hours of help a day, and mom hasn't exhibited any dangerous behaviors, this can be a good solution. As for the falling, I'm sure you realize that mom will fall wherever she lives. As you've pointed out, it's the getting back up that's now the issue. Does their community offer a pendant system? That could do the trick there. They both could wear them. There are even pendants that have accelerometers in them so they can recognize if the wearer has fallen.
You may find that the director has them on her/his radar. They may already be getting some help informally (like the staff responding when your mom has fallen). When that starts to occur, they'll start tracking their needs and at some point you'll get a request for a sit-down.
I agree with the previous poster that their doctor should be consulted. However, keep in mind that s/he only sees them briefly during visits (and elders are often putting on their best face for the doctor). The dementia spectrum is so broad and nuanced that you really need to live with someone to see how they are. A doctor may test for their 'level', but that doesn't begin to give a real picture of how a person lives.
Assisted Living is great when it's needed, but if they were my parents, I'd want them in the least restrictive environment possible. If they like where they live, why not stay if they can?
Hope all goes well.

I think that you know the answer. I also think that it is time. Good luck. We have all been there, in some sense.

Get their doctor to order them into a nursing home.

my father is only 69, and due to his dementia the DOCTOR ordered him into the nursing home also.