I'm a cyclist on the side of people being allowed to ride on trails in National Parks. The arguments for are simple: allow bikes so cyclists can enjoy nature while doing a healthy activity they also enjoy. Bikes are faster than hiking, allowing you to explore more trail in a day, make the descents a lot easier, and also enable older people who can't hike well (bad knees) to still enjoy a gentle trail by cycling.
The arguments against are mainly related to trail maintenance and safety. Bikes can make a mess out of muddy hiking trails, especially when strong cyclists try to pedal through the mud and dig up ruts, but maintenance wise that the only con to a bike, unless you want to include tire tracks on rocks. The main concern is safety; people bombing down a popular trail and colliding with hikers, or scaring horses, which usually results in people getting bucked off and hurt. Parks are concerned about liability. Mountain Biking is a high speed sport, and high speed sports are all capable of producing serious injuries. Park officials don't want to have to deal with crashes and wipeouts, or complaints from jumpy hikers about that crazy cyclist who passed them going "dangerously fast".
Parks are all about keeping the peace, which mostly means the peace and quiet in the administration building, they don't want to deal with paperwork, which incidents of any type create for them. It's easier to ban an activity then it is to deal with the injuries and drama from it.