You won't be able to "see" any of the parks at that pace--you'll be able to glance at them as you whiz by, maybe. I strongly suggest spending all of your time in one area (Montana/Wyoming) or another (Utah/Arizona). You'll get more out of your trip, you'll spend less time driving (and driving and driving and driving and driving), and you'll actually get to know the parks.
Each area has its own positives that time of year. The desert is starting to cool down (and I would NEVER go there in summer--been there, hated that), while in Yellowstone, the wildlife are really active and the tourist crush has abated.
If there's one common thread in all these various discussions of "I want to go on this big trip," it's that the contemplated trip is just about always too big. It's MUCH more satisfying when you explore a few places in depth than when you zoom from place to place. Either half of your trip could easily keep you busy for weeks. It's better that way, and importantly, it'll actually feel like a vacation.
