Hello, 

Thank you for your question! Bryce Canyon does allow horses brought into their parks to ride, however you must hire a guide to ride along with you. You can read more about that at this link: https://www.nationalparkstraveler.org/2014/03/you-can-ride-your-own-horse-bryce-canyon-national-park-you-have-hire-guide24806

As for Zion and Paria Canyons, they do offer horseback riding tours, but you would need to call in advance to bring in your own horses. You can call specific ranches in advance to see if they will allow your own stock onto the trails. You can get that Southern Utah ranch info here: http://www.visitsouthernutah.com/Horseback-Riding

Best,

Sarah