Most people w/ bully breed dogs will tell you to avoid dog parks...basically because of the possibility of dog aggression bully types are susceptible to. If your dog should get into a fight w/ a non bully for whatever reason, you will be blamed and your dog will suffer the consequences. There's a current 'Suffolk' story in the news, you may have read about it... a 'pit bull' was just euthanized for human aggression...turns out the dog didn't even bite the 'victim', but he was a 'pit bull'..therefor guilty. better be safe than sorry.
The other issue is...you have a puppy...you don't know if other peoples dogs are carrying diseases, and puppies are more at risk than adults...parvo, an often fatal disease, for example, is very common on LI. It's a virus and can be spread in such ways as through a sick dogs poop. The virus can live in the ground for many years.
If you don't know anyone with a well cared for dog you can socialize your pup with...and If you're willing to travel to Brooklyn, or Manhattan...there is a pit bull dog club which meets regularly in parks on weekends with dogs. Its called the 'pit bull / pit bull mix rescue group' and you can join online through 'meetup', just google meetup.