We carry fairly extensive liability insurance because of our studio, but have only been asked about it when discussing the need of permits for doing outdoor portraits in parks. I haven't had that discussion in California state or local parks, but when the topic has come up in other states liability insurance was expressed more as "a good idea" than a requirement for doing the portraits. I had established good rapport with the park staff over the course of the conversation, and they seemed to be expressing more of an individual concern versus a policy issue. They simply hoped that if someone got hurt while I was photographing them, the park might not be held liable. My having the insurance seemed to reassure them that there would be at least a modicum of alternatives to court in a worst-case scenario.
The real virtue in having the insurance has been in exempting us from "letter of the law" treatment, so far. I say that because in most parks with which I am familiar the dividing line between needing and not needing a permit is the use of a model, a product (as in product shots), or a photo crew. They regarded a permit to allow me (without a crew) to take individual or family portraits as overkill, something beyond the real intent of the permit process. Park staff said they consistently have the most problems with television or movie crews or product shots, and occasionally with photo classes or workshops. Based upon past atrocities, for those types of shoots they want a schedule of activities and will often asign a park ranger to accompany the crew at all times.
As a professional or at least someone using "professional-looking" gear, if I anticipated permit issues as appears likely in some California parks, I would make the park headquarters my first stop as a matter of course. I might have to buy the permit and show my proof of insurance, but I doubt it- In my experience the staff are anxious about the activities of photographers based upon past abuses and are not really concerned about reasonable activities of individuals, especially those that care enough to make an appearance at the park headquarters.
A friendly and professional approach to them for information and a willing attitude on my part has gone miles to reassure such nervous caretakers, who are in fact professionals in their own rights charged with protecting their domain. I hope that doesn't change, but it won't take too many incidents such as the fires in Arches to change the landscape for all of us. My rule of thumb is "If in doubt, make an appearance and smile."