There isn’t actually mountain biking in Bryce or Zion National Parks, but there is plenty of super-classic riding just outside the parks. We used Escape Adventures, a qualify outfitter that runs van-supported trips. Basically, they have linked together great single track riding usually ending at a campsite and food.
Our kids were pretty young to do this at 9 and 11 years old. We practiced at home by doing Rock Creek near Mammoth and The Flume Trail near Lake Tahoe. It worked out well. They had a great attitude and worked hard to get too far behind other riders on the trip. We (my wife and I) also worked hard by sometimes helping them up steep hills by dropping our bikes at the top and going back down to bring up their bikes. It was worth it.
The trip starts in St. George so that is where we flew into. It is a fun place to fly since the runway is up on a table-top mesa (at least for now, there are plans to move it). We shipped our bikes out ahead of time, although you could just as easily rent from Escape. I guess we like our bikes.