It's almost assuredly a liability issue. Every couple of years our insurance provider sends a risk assessment specialist to evaluate our operation from a liability standpoint. I can easily see where riding a bicycle in a park is not the safest thing in the world. Kids are forever darting out in front of 45' motorhomes who's drivers are looking for their site, not kids on bikes. It wouldn't surprise me if one of those risk specialists told us we needed to have bicycle rules and policies.
Then again, maybe the park had someone who crashed their bike and suffered a serious injury to their head. Perhaps they just decided they only wanted to clean up someone's brain matter once in their lifetime. We have had medical emergencies and people suffering injuries on our property and it sticks with you. You don't just laugh it off and move on. If a helmet rule would prevent us having to watch an ambulance transport a child to the hospital, it would be worth any inconvenience it causes those who oppose such rules.