I frequently ride in Bryce Canyon, Capital Reef, Grand StairCase Esclantee, Zions and Yellowstone. So yes you can ride in the National Parls, But the rules differ a little with each park.
Bryce Canyon is small park, You can easily ride the entire trail system in one day. They have a concessionair that has paid for the rights to take tourist on trail rides in the park. So their rules protect him. Since most of the trails in Bryce are often on steep switchbacks and you are not allowed to leae the trail system. And it's assume that horse owners are better riders than tourist hiring a guide. You as the private owner always have to yeild to the the guide service. They don't want to find a little Japanese grandma on horse following your horses back to the trail head while the rest of the tour group goes in the opposite direction. Since you can't pass and can't leave the trail. Sometimes this means you have to turn around and ride back to the last cross trail where you can get off the trail and allow them to pass. We always check with the guide and find out what time they run their trail rides and time our rides appropriately so we don't ever meet them head on on the trails.
In general National Parks require you horse to have a Coggins test ( most require that you carry that on the horse so a back country ranger could check it in the field) Trailers usually have to be swept clean. No manure on the floors when you enter the park. You are suppoed to feed Certified Weed Free hay for 3 days prior to entering the park. And Yellowstone in particular doesn't want any hay in or on the trailer. They are very concerned about non-native species getting transplanted into the park. But processed feeds are OK. ie hay pellets where the grinding process to make the pellets destroys seeds
Both Bryce and Zions have a very limited trail system. They are beautiful ride. Both areas have extensive trails outside the parks that easy to access. So when I go to Bryce, I ride one day in the park and several days just outside the park.
Yellowstone on the other hand is a VERY large area. You could ride for weeks and not repeat any of the trails.
Most of the parks do NOT have any camp grounds for horse camping. Yellowstone does allow back country camping with a permit. So most of the time we are primative camping somewhere outside the park and either day tripping into the park or doing a pack trip in.
Feel free to ask more questions.
Red Mountain just outside of Bryce
Grand Staircase Escalante