OK I have foundsome info on carrying in state parks:
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.
The link to this page is bellow and states "Concealed firearms permit holders are exempt from this prohibition" but I can not find a valid reference for this statement in any law.
Finally, Utah Admin. Code § R651-612-1 generally prohibits the use or possession of firearms in any park in the Utah park system, unless the weapon is unloaded and encased. Concealed firearms permit holders are exempt from this prohibition. Id.
(Found about 1/2 down on this page)