It's very generous of someone to offer to take in an aging parent. Unfortunately it's not always that simple.
Many older people can be at risk for "geriatric syndromes". Falls is a good example. Does your home have accessible, stair-free entrances? Do you have good lighting? Do you have a bathroom on the main floor? Is it easy to get in and out of the bathtub or shower? Do you have bedrooms large enough to accommodate a commode and possibly a mechanical lift? If any of these is a "no", you'll have to either modify your home or move.
Many older people have problems with urinary incontinence. Are you prepared to change your parents' undergarments? To wash them down there at least twice a day? To give them a bath in bed or in the tub?
What if your parent develops dementia? Often they can mix up days and nights. You have to be constantly alert, especially at night, as they could become confused and wander off, even leaving the house. They could become aggressive. I have been hit, pinched, kicked, scratched, screamed at, and sworn at by residents who had dementia or other mental health issues. Are you prepared to deal with that?
You'll also have to make arrangements for respite care periodically. You need a break and this is very important. This means you'll have to either put your loved one in a respite bed or hire a respite carer for them periodically.
Do those who are convinced caring for parents at home is the best option see what level of care that may entail?
Now, I'm not against good home care. Some people are able to stay at home for a very long time. My grandfather died at home. But others aren't that fortunate. If their care needs exceed what can be provided at home, they'll need to move to a facility.