Public transportation in Switzerland is so good that I'm not sure a car would add substantially to one's flexibility, particularly for a trip of this length. In contrast, having a car would add to some complications -- finding parking, navigating, etc. And the driver doesn't have the opportunity to savor the scenery in the same way that a train would allow. JMO.

Fantastic advice above
One base. (.)
https://www.sbb.ch/en/travel-informa...p-support.html is worth digging into because there are native Swiss who cannot move easily there are sensible services to help them.
I would not hire a car, I would limit my movement.
Strict Veggie, I would use an app to ensure there are sufficent restaurants near your hotel to offer variety. Veggie will not be an issue but distance to a good veggie restaurant might be, especially after a hard day.
I would stay in a hotel incase you need medical services
I would assume you are going to lose at least one day you were not prepared for, my super tough father of that age, just needed to chill for a day when it all got too much.
Altitude, as janisj said, would worry me ( I recently blacked out in the alps so tell-me-about-it)
Melnq8 is spot on, where to go.

Lucerne might be a good base

AS I said upthread if you have Indian passports you will need Schengen visas, and you need to look into what is needed for those in plenty of time. Since you are planning to travel next month you should be doing that now, and make sure anything you book is refundable in case the visas don't come through in time.

After post 16, everything changes.
In this case, I recommend to stay at Montreux all the time.
There are tons of day trip possibilities from Montreux

My suggestion...immediately, today, make a reservation for your entire stay at the Grand Hotel Suisse Majestic in Montreux. Sounds expensive but it's a Marriott, and very good. Arrive by train and walk them across the street perhaps 50 yards from the rail station to your hotel. Unpack their voluminous luggage and remain unpacked for your stay. Get up every morning and drink coffee on your balcony looking at the beautiful lake. Leisurely decide what to do for the day...a slow walk along the prominade, a cruise on a steamer, a trip to France, to the mountains. The temperatures will be moderate and the cost will be less than, say, St. Moritz. You will have access to most "scenery" typical of Switzerland without battling crowds and rough terrain...walk across the street to the train station and go wherever you'd like and return to your hotel and an evening meal. With elderly and sometimes stubborn parents I suggest you work around their limitations and eccentricities and kind of take control ..stay in one place, do day trips, have more fun than running them to death. You only have a few weeks, decide quickly!

i ageee with wildiowa’s advice.

IIRC, the lakeside promenade in Montreux is wheelchair friendly, wide and level, with plenty of benches where one can rest and enjoy the scenery.
On a clear day, consider the 50-minute cogwheel train to Rochers de Naye. Yes, on a clear day. When we did this years ago, by the time we reached the top, we were quite literally in a cloud and the views were completely obscured.

Advice from wildiowa is excellent.

SBB (Swiss trains) offers assistance for travelers with disabilities.

https://www.sbb.ch/en/help-and-conta...-mobility.html

And they’ll deliver your luggage for you, too.

https://www.sbb.ch/en/tickets-offers...-services.html

I should clarify that the bases are :

11 (reset day), 12 and 13th night: Geneva
14, 15th night: Lucern
16, 17, 18, 19 : Gündlischwand

So we have 3 bases

Honestly, this plan sounds awful to me. Geneva makes no sense given your stated goals. One full day in Lucerne won't give you time there to enjoy what it offers. And you've now decided to go somewhere in Austria instead of staying in Switzerland?

Your trip, your call, but why you haven't accepted the suggestion to spend time in Montreux is befuddling.

I think it is good of you to try to plan a trip that will meet your parents' interests and needs. Unfortunately, I don't think you are succeeding.

I see: >>Gündlischwand, a municipality in the Bernese Oberland, offers access to the natural wonders and landmarks of the Jungfrau Region. Situated along the Black Lütschine river, the area serves as a base for exploring mountain experiences and cultural sites.

I can see the attraction. However, as a person in your mother’s age range, with mobility challenges, I would opt for Montreux over Geneva and Lucerne and maybe Gündlischwand too. There’s a direct train from the Geneva airport to Montreux. Use the services available from SBB (see my post #28). Relax and reset after your long journeys. Let the folks at the hotel help you find restaurants and make suggestions and arrangements for daytrips.

Thanks, k_marie -- that didn't show up when I googled it.

I, too, can see the appeal. But for this trip? I don't think it's a good choice.

I keep wondering why you haven't taken our recommendation to stay in Montreux. In post # 25, wildiowa gave you information about how very easy it can be ... only a few short steps from train to hotel, etc. Maybe you don't have a good picture in your mind of what Montreux is like? Here are some photos I took at the very end of May 2013:

Night time view from my hotel:
https://cimg5.ibsrv.net/gimg/www.fod...1a9e97e80.jpeg

View of the promenade:
https://cimg6.ibsrv.net/gimg/www.fod...101e4e571.jpeg

View from one of the steamships plying the lake:
https://cimg7.ibsrv.net/gimg/www.fod...2645d7a06.jpeg


One of the harbors along the lake, seen from the same steamship:
https://cimg9.ibsrv.net/gimg/www.fod...56b497333.jpeg

The Lavaux Vineyards, from that same steamship:
https://cimg2.ibsrv.net/gimg/www.fod...779487513.jpeg

A square in Lutry, an easy train or boat ride from Montreux:
https://cimg1.ibsrv.net/gimg/www.fod...1b1a9f1f5.jpeg

View from the Rocher de Naye, an easy (and interesting!) cogwheel train ride from Montreux:
https://cimg6.ibsrv.net/gimg/www.fod...3ce78d92e.jpeg

A street in Gstaad, an easy train ride from Montreux:
https://cimg9.ibsrv.net/gimg/www.fod...82b77cd54.jpeg

SOoooo many other options....

https://cimg8.ibsrv.net/gimg/www.fod...5ae82425d6.jpg

One suggestion for elderly parents IF you include any cities like Basel, Bern, Thun, St. Gallen, etc in your trip: a velo (bicycle) taxi tour. I arranged these for my mother for her visits to Basel (where I live) , Dresden, Lugano, and other destinations. They are great options for older, less mobile visitors. The velo taxis easily hold two people. The drivers are friendly and professional (most of them use e-bikes these days so they're not being overworked). They can go into many pedestrian-only areas that are off limits to cars, buses, trams etc. They will also happily comply if your parents want to stop and take some photos.

I started doing these with my mother when she was in her mid 80s and continued doing them through her last trip to Europe at age 91. We both really enjoyed them.

WeisserTee, that is a great suggestion! I have never taken one of those rides as they looked so awful for the drivers but I can see that e-bikes are a game changer.

There's also an elevator steps from the Montreux train station that goes to the promenade level, where one can access other hotels and restaurants.

When we were in Montreux a few months ago, the elevators were OOO. Maybe it was the inspection period. General rec: Restaurant 1820 is a great spot for lunch (reservations strongly recommended)