Do you need a permit to have a birthday party

No, not to my knowledge. There are public bathrooms with tables that are opened to the public.

It depends on if you are wanting to have a party in the public park or at the clubhouse/recreation center. The clubhouse/recreation center may require a permit if alcohol is going to be served. The public park is within a close radius to private homes so you should contact the Parks and Recreation center directly to be on the safe side.