The default is that you need one
Californians repeat that garage doors are exempt. The code does not say that. §R105.2 lists what is exempt from permit — fences under a height, retaining walls under a height, sidewalks, paint — and overhead or garage doors are not on it. That makes a door replacement presumptively permit-required under the base code, with any leniency coming from local practice rather than from the code text.