R.K.
It is a public park and there for everyone to use. However if the person hosting the party has rented the space, for instance the pavilion, you need to respect that. This doesn't mean you can't use the restrooms but you shouldn't go in and sit at the tables or try to get involved in the party games and activities.
As far as the swings and slides etc go I do not think you can rent those areas and everyone should be able to play on them.