The victim of a personal injury or injury to the victim's dog probably will not prevail on a claim against the local government entity that established the dog park. Generally, there are immunities that protect government entities from many claims; if the entity believed that it was doing something beneficial for the community, it is hard to get around the immunity.
However, failure to enforce its own rules and regulations may result in governmental liability. Dog parks frequently are governed by special rules that are either posted or part of the local municipal code. If an irresponsible dog owner has a habit of breaking those rules, and the local animal control officers do nothing despite being informed, a person who sustains personal injuries or injuries to his or her dog may prevail against the municipality.