Not that riding a bike is known to treat parkinson's. It's that being able to ride allows exercise and encourages a positive outlook.
There are diseases treatable by cycling. ADD comes to mind. Depression and some bipolar disorders. Addictions.
Perhaps instead of rooms and beds, mental hospitals and substance abuse rehab facilities ought to have bicycles and cycling paths.
Seniors get problems with balance. Which is one reason T'ai Chi is recommended. Maybe cycling ought to be too, but the problem is that the very challenge of balancing on a bicycle makes riding it dangerous to someone with impaired balance and fragile bones.
There ought to be a design for a stationary bike that mimics the tilt of a cornering bike. Not sure how this would be possible unless it were mounted on a merry go round.