I've written several posts about our attempt to bring our dependent parents to Switzerland from the US. Rather than repeating those here, you can search for them, probably in the 'family and health' section.
Quick summary:
We live in SZ, which is generally thought to be more skeptical of chain immigration. I have no idea how Basel bureaucrats would react.
I bring this up because my understanding was that IF - and this is a huge 'if' - you were to get something other than an outright 'no', a situation such as ours/yours might be handled on an individual basis, and might comes down to whether or not a case that can be made for compassionate consideration. And of course, you'd need to be wealthy enough to pay for your parent's care on your own.
So think about how you would present your case.
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As Medea point out, there is very little chance under regular immigration rules. As a non-EU non Swiss you have no right to bring parents here.
Our first attempt went nowhere, because both parents were still alive, albeit incapacitated. The assumption was that two people can care for one another, so no compassionate case could be made. After MIL died we tried again to bring FIL here.
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The assumption from local Powers That Be was that we should go home to care for our parents. I might guess that many would feel similar about non-EU folks, so be prepared.
You'd need to think about how to argue that one. Do you contribute to Switzerland to such an extent to make a compelling case that it is in Switzerland's interest for you to stay here, and bringing your parent here is needed in order for you to stay? Are you a business leader, job creator, wealthy, or fall in some other category where Switzerland tends to roll out the red carpet?
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Key in speaking to the Gemeinde was finances. We had to show that we had the assets, now, to care for FIL for the expected rest of his life, showing that money from Swiss coffers would not be needed.
And yes, it was a lot of money.
Broadly speaking Swiss immigration policy does not want people coming here who have not contributed to the country yet cost the country money.
Our attempt was 10 years ago. At the time even though we expected FIL to live with us we drew up a budget assuming nursing home care.
Some rough updated figures, based on what friend are paying monthly for dementia care:
Dementia ward in local Gemeinde (public) nursing home: 8K for the room, plus around 2K for daily care. In a private nursing home, 12K for the room, another 3K for daily care.
A note here: The public nursing home costs count in contributions from insurance and the Gemeinde. This could be a sticky point in your argument, as your parent could be seen as taking from the Swiss taxpayer, without ever having contributed.
Were it me, I would base the budget on private care only, to avoid that perception.
Another issue, perhaps of growing importance:
There are not enough nursing home places in Switzerland to meet demand, the shortage is acute in dementia care. Many nursing homes have a wait list. By bringing your parent here, you could be seen as taking a place from a Swiss citizen who has been waiting for the room. If you base your argument assuming you would use private care only, financed by you, at least there isn't the perception of taking advantage of local citizens.
Again, I make this recommendation based on my experience in SZ I have no idea if other cantons/communities would feel the same way. You might face completely different attitudes.
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What this boils down to is - go ahead and try, but be prepared for a 'no'. Make a compelling case as to why bringing your parent here is necessary.
It's a pretty steep uphill climb.
We never got an answer to our application, as FIL died before a decision was made. We withdrew the application.
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An alternative consideration:
The money we budgeted for elder care here went a long way to providing care for the ILs in the US. During the worst of those years, one of us traveled back and forth every few weeks. It's a hellish juggling act, but it can be done.
Dementia care means that someone needs to be on hand to advocate for the patient, even in the best of facilities. Do you have anyone trustworthy you could ask (and perhaps pay) to take on that roll for the times you can't be there?
Or a friend's solution was that the employed spouse stayed in Switzerland, the other spouse went back to the US for 3 months at a time, making sure to remain within the boundaries of her permit. Again, hellish on the couple, but doable if you are both on the same page.
We had help from ILs' neighbors and members of ILs' church. If your parent was active in a church community you might find help there, as many have extensive outreach programs.
And as an FYI, if by chance your parent is in the Chicagoland area, I can recommend an outstanding memory care facility, the one we ultimately used for my father. (Yes, we were dealing with three parents' needs at the same time.) FYI, we did not consider bringing my father here, as there was no way in his state of dementia he would be allowed on a plane.
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These are extremely difficult times, I know. You have my sympathy. I hope you can find a solution that works for you and your family, and your parent.
Wishing you all the best.