I don't think you realize how large Utah is... the parks are scattered over a very large portion of the state, and staying at one town to see them all is probably impossible.
We take the mh to different towns (rv parks) to visit "each" of the national parks as each park requires multiple days to explore all of it, so there is no time to be driving from one town to see a park in another part of the state.
Use Google Maps to locate the nearest town to each of the parks and then locate an rv park in that particular town. Since the parks are heavily visited by rv'ers, there is no shortage of rv parks to stay at.
Good luck
Ron