California state law requires a work permit from the high school or equivalent office, and a minimum age of 16 and a half from what I remember. That said - most positions in theme parks require a minimum age of 18 to meet legal requirements for operating machinery, chemicals, and ride operating systems. This would eliminate most positions for you, unless you were to go into a department such as Amazing Photos, or retail sales, which don't require much in the way of adult consent for potentially hazardous equipment/chemicals. Even then, they may have different requirements for age that would keep them from wanting to hire a minor.
TL;DR: Believe the staff. Check the USH Jobs website for seasonal position requirements, and keep your eyes on it!