I thought I found the answer in California Firearms Laws Summary :
"Unless otherwise unlawful, any person over the age of 18 who is not prohibited from possessing firearms may have a loaded or unloaded firearm at his or her place of residence, temporary residence, campsite or on private property owned or lawfully possessed by the person."
However, the same section has this note:
"If a person’s place of business, residence, temporary residence, campsite or private property is located within an area where possession of a firearm is prohibited by local or federal laws, such laws would prevail."
California gun laws are, generally speaking, very restrictive, and I don't know whether any "local or federal laws" would prohibit a person without a CCW permit to bring it to a campsite in a county park.
The specific park I'm interested in is the Sanborn park in Santa Clara County.