It's very busy in YNP in the summer and parking will be a challenge no matter what type of vehicle you are driving. The bigger the vehicle the more difficult it will be to park at the many thermal features. No problem at Grant, Old Faithful or Mammoth.
As far as roadside camping, that is not allowed inside the park. Do you have reservations at a campground in or out of the park? If not you should make some ASAP and consider yourself lucky to get a space. There are a number of first come first served CG's in the park but all CG's report full by 8 to 8:30 AM. I think the idea of renting a car in W. Yellowstone is a good one. There are many CG's in town and if you don't mind an additional 15 minute drive you can stay at Valley View RV park in Island Park ID.
