Finding accommodations in the national parks at this point will be impossible. I would recommend either a detour through Yellowstone or through Salt Lake City and the various Utah parks such as Bryce, Canyonlands, Moab, etc.
On the way to Yellowstone, Three Forks and Big Sky are both cool places to visit. Idaho Falls or Pocatello are both decent stops on the way to Salt Lake if you don't want to do the 12 hour drive in one shot. If you go through Utah check out highway 65 just east of Grand Junction. I haven't been but it looks like a fun drive.
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