A couple of time to note: There is a teachers convention around the third weekend in October (Oct 20-24) and with a four day weekend, the parks fill. In Moab, avoid the week before Easter as that is the Jeep Safari and way too many people in town and around the area to make it much fun.
Spring and fall are popular times in southern Utah as that is the best weather. Even late August can be quite warm, as can late May. But with planning, you can find places to stay. By mid-March, Zion and Moab are fine, but Bryce can still be snowed in.