Just outside of Bryce Canyon National Park is a Utah state park called Red Canyon. The hoodoos aren’t as dramatic as in Bryce but has a similar landscape and the advantage of being dog friendly, as are most, if not all, of Utah’s state parks.
Front the visitor’s center you can start a 5 mile loop hike along Golden Wall which will drop you into the campground. Much of the hike is exposed so in the summer have plenty of water for you and your furry friends.