Most Public Parks have, somewhere near the Entrance, a Notice Board giving details of what can or cannot be done in that Park (Local Bylaws). The majority that I've looked at usually include; 'Dogs must be kept on a Lead at all times, and must not be allowed to cause a nuisance to other users' (or similar wording). And yes, I am aware that it is usually ignored.
If your insurance company checked, remember they don't like paying out, there is a possibility your policy could be void, and if the dispute became
heated, there is a small chance you could be charged with breaking those Bylaws.