we have always preferred late August to mid Sept for the Yellowstone/Grand Teton area, but part of the reason is that we're from south Texas so we're seeking high mountain country with less heat. Both parks are very popular so there will still be lots of people, but not quite as many as in July. But reservations are totally necessary if you want to stay inside Yellowstone, and it's best if you can do that because the park is so big. We like Bridge Bay because it's sort of Centrally located, and we enjoy dining out over at the Lake Lodge cafeteria. Fishing Bridge is also well located and has full hookups (but more expensive).
Make reservations as far ahead as you possibly can.