They don't quote the whole law... Much like how the airport was. It is Legal to carry in a state park, just not shoot. 53-5-701 refers to utahs state laws (76-10-500+)R651-612-1. Weapons Prohibited.
Possession or use of firearms, including air and gas powered types, traps and all other devices capable of launching a projectile which could immobilize, injure, or kill any person or animal or damage property are prohibited in the park system unless:
(1) The weapon or device is unloaded and cased or otherwise packed away to prevent its use in the park area.
(2) The weapon or device is being used for the legal pursuit of wildlife, see R651-614, or in accordance with UCA 53-5-701 Concealed Weapons Act.
(3) The weapon or device is being used by authorized enforcement officers in the performance of their official duties.