Hi, we were in Yellowstone in July of this year (entered through the north entrance). While we got to see everything we wanted with one exception, the park was very crowded. Parking is definitely st a premium, and i couldnt imagine trying to get a spot with anything other than a small vehicle. The larger spaces are there, just limited.
The last previous time I had been there was about 30 years ago. It was quite how you described it. We tent camped and got sites with no reservations. Not so much anymore!
Id suggest renting a car so you make sure you can see everything you want.