So, does a park that forbids firearms have a responsibility to protect me while I'm in that park?
That would be nice, but I highly doubt it.
In 2005, the Supreme Court decided again that police have no legal duty to protect an individual. (See Castle Rock v. Gonzalez.) In practical terms, an individual could not successfully sue the local police department for failing to respond effectively to a 911 call. Even if a state or municipality has restrictive gun laws, police have no legal duty to protect an individual. Thus, it would be highly unlikely for park police to be legally responsible for an individual's protection.
If more people fully understood the way American laws work with respect to police protection of an individual, a strong majority of people in America would be pro-gun for law abiding citizens.