We looked into this a while back, and are doing so again now following the latest crisis - but the stumbling blocks for us are permits, health care, housing and ability to integrate into the new community. My in-laws are in their late 80s and in poor health; they need assistance to do most things. They do not speak German, only English.
As we are non-EU the permit issue makes it a non-starter for us. If your parent is EU that's a whole different story. I don't know much about EU permits, so I'll let others address that.
But some other things that made us think twice:
As others have pointed out, make sure that your parent's home coverage is extendable here (some are not), is accepted here, and is sufficient considering Swiss prices.
But also think about how different the Swiss system is from the one your parent is used to. Would your parent be able figure out the ins and outs of the Swiss system in order to get the treatment she needs, could she be her own advocate if necessary?
We worried about this - the ILs are used to a very different health care system and would have trouble navigating their way through this one - and not just because of the language barrier. While I could find them an English-speaking doctor, their expectations of the doctor/patient relationship are quite different than what is typical here. And at a more basic level, they would be unable to communicate with nurses and care assistants - all of this means that their health care needs probably would not be met.
Given that they have gold-standard health care in the US, it makes no sense to bring them here from that standpoint.
Housing: Is your housing adequate to your parent's abilities and needs? If not, can you find something that is?
Housing is another problem as my ILs cannot climb stairs nor walk more than a few meters. We looked into making the house handicapped accessible, but the neighbors told us that they would oppose our permit application. That left us with selling our house and finding something accessible for them - but a large single family home with level access where one could drive directly to the door... this simply does not exist here.
Finding them a flat in close proximity to us was out of the question as the whole purpose of the move would be to be under the same roof, so that we would be on the spot to help them.
Integration: Do you think your parent can cope with a new culture and community at this point? Would that be important to her to be able to do so? Could you cope if she could not adjust to the differences?
How can I put this... The ILs are very, very American
and very proud of it. They simply cannot conceive of living another way, and at their ages they are not going to change. Adjusting to living in a communal fashion as one must in Switzerland (living in a villa or in a flat, no man is an island here) would be a nightmare for all.
Added to that they a bit doddery and at times can be curmudgeonly. At home they are known, loved, and respected for all they have done for the community over the years - people look up to them, and look out for them, help them through the occasional 'senior moment'. Here they would have no history, likely would simply be viewed as 'stupid old foreigners' and I fear treated with a degree of contempt. This is not how one should spend one's twilight years.
Bottom line, I don't think the ILs could cope with the change. They would have us, but they would lose much that is meaningful to them. It would be unfair to them to attempt a move at this point. We need to provide for them in their own home and community.
It's a tough situation, I know. I applaud you for looking for ways to make this work for all concerned - and wish you the very best.