Are you still going to be traveling through Colorado at all still?
As others have said Yellowstone, while an incredible place, is quite crowded and busy this time of year. Grand Teton is amazing, but will be busy too. I'll be traveling through there to (hopefully) climb the Grand later this month (weather depending).
No matter what National Park you go to, as long as you are willing to hike a decent distance the amount of people drop dramatically. Something like 95% of all park visitors never get more than a 1/4 of a mile off pavement. Camping and driving could be an issue but if you have any questions on free Forest Service/BLM camping you're more than welcome to ask me or check out
Free Camping Near You | Go Camping for Free!
The Wind River's are about as remote as it gets in the USA. Beautiful landscape and incredible mountains. Totally worth looking up if you're willing to do a good backpacking trip there.
Depending on how committed to a big drive you could also consider Glacier National Park too. Much less crowded and somewhere I will also be visiting later this month on my vanlife roadtrip.