What to do if family caregivers refuse to put the parent in assisted living when the health is declining?

Are they planning on in home hospice? Have you spoken to them about types of care? Can you give us more information please?

If a persons care or lack of care is causing you concern you can contact APS where they live and report them as a vulnerable adult. Realize that what you think as negligence may not be according to the authorities. But at least you will have brought the person to the attention of someone who can intervene if the circumstances warrant.

Is the Health declining to Bad Caregiving? Or is it just the natural order of things that happen to elder people as they get older? Sometimes it is difficult to read which one it is, perhaps the LO isn't able to get out of bed herself, etc.
Perhaps the pople who are taking care of her, and want her to be in a well fitted home, they are willing to take on the responsibility. Maybe you can offer a few hours of your services to help them out..That is if you are not too far away...

Location Location Location is the most important part of a place to put mom. I can drop in any time to visit with any other arraingment

We really need more information to give you helpful answers.

Do the family caregivers have PoA or guardianship of the parents?
Do the parents have cognitive issues? Or just medical ones? What medical issues are they having that is causing concern?

If the parents have all their mental faculties, they can make their own decisions about where they go, even against the caregivers' opinion -- they cannot be forced.