Yes. Probably not in the most popular parks this time of year (Zion, Bryce, Arches), but maybe at some of the more remote ones like Capitol Reef.
You should be able to find a tent site in an independent campground in one of the little towns down the road a way from the parks. Just stay a little ways away, get up early and drive to the park for the day.
Also, don't forget there are some nice state parks down there too. If you're more in the middle, check out
Goblin Valley State Park.