I guess the question should be, if the park is owned by a "corporation" and all land owners are "members" of the corporation, does that not entitle the landowners to use the corporation property as owners? Unless specifically prevented in the by-laws? The POA doesn't have any rules against off leash animals in the deed restrictions or by-laws.
What about if I go to my buddies house and let my dog off the leash? That isn't my property but I have consent to allow the dog to be off leash.
Obviously law still applies and that law is that I cannot take my dog off my property without a leash/restraint. I believe that the private park is my property along with all other residents. Example, you cannot be drunk in public, but you can be drunk on private property unless prohibited by the property owner.
The more I looked into this, essentially no law enforcement can enter the property without a warrant or PC. There in lies the problem, PC will be given by some jack-wagon who is mad I am throwing a ball to a dog 300 yards from anyone else.