My dad is diagnosed with Parkinson's and lives in India. Are there any at-home exercise machines/vibration that one can recommend?

Go to the Parki sons Assn website to familiarize yourself with the diseaseand join an in person or web based support group.
And yes there are exercises such as boxing, he can do but he really needs to be motivated.

So, Mom and sister live with you in the US? Dad remained in India?

I read where bicycling is a good exercise for Parkinson people. Not sure if a stationary bike would work since I think it has to do with balance.

If sister is working against you, why do you let her live with you? You need to keep a clear head and do your best for your dad, but that can't likely happen if a hostile person is living in your space. Try to find a senior citizen helping place in India near your dad. Find out if they have any good ideas about taking care of him long distance. And don't tell your sister.

The disease WILL progress, that's one certain thing about PD.

Medications and keeping up with moving as best you can are the best PD 'cures' (There's no absolute cure, as you know)

Do you live in India? Sister? How is she manipulating him? Is it possible for you to go to HIM and accompany HIM to drs appts to see what's available?

I have a sweet neighbow who is 80+. He is amazing, just amazing. He had a deep brain stimulation 15 years ago and his wife said he was almost like baseline--it really worked for him. He does take meds and he is as active as possible. His wife says he is on the downhill slide now, but they got 15+ years of almost no-symptom years. Sure wish this had been available for my sweet daddy.

Curious as to why you're worrying about vacation time?

Consider a portable pedaler that Dad can use when seated in a chair.