When I started to read your question and saw the words "limited mobility" my first thought was "wheelchair or EVC." And then I read the second part of your question. Yikes. I'm sorry to tell you that the in-park transportation options are extremely limited. There is the WDW Railroad that circles the Magic Kingdom and the Friendship boats that go from one side of Epcot's World Showcase to the other, and that's about it.
Are you sure you can't convince your parents to rent EVCs? A Disney vacation is VERY intense with walking. Most people find that they walk anywhere from five to eight miles per day at WDW. Disney Mom Laura once wore a pedometer at WDW and found that she walked 12 miles in a single day. This can be quite a toll on a body.
If you'd like more information on mobility issues at WDW, you should look for is a book called PassPorter's Open
Mouse for Walt Disney World. This is the definitive guide for anyone
visiting WDW with any sort of physical issue. The potential challenges
with each attraction are described in great detail. You can get this
book at Amazon.com or from major booksellers. You may also want to stop by the Guest Relations office at the parks and ask them for a free copy of the
Guests with Disabilities Guidebook for
that park. You may also inquire about getting a Guest Assistance pass
that will allow you easier access to some attractions.
Good luck with your planning.