Typically you can throw a party up to a certain number of people without a permit. 50 or 20, I can't remember. There are signs stating the rules and possibly the contact info for reservations.
People usually just set up the party and no one says anything. I've never heard of reserving space there and many people do things like parties or barbeques. It is a public space after all
There are reserved areas right next to the play ground but the green open area is up for grabs.
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