Possibly , you'll have to call/ email the offices & confirm. It would most likely have to be at one of the eateries or taverns on the grounds. I don't know if they have any designated areas for private parties. I know they do weddings there.
Yes, you can have birthday parties there. You can handle it yourself as long as it's in a public area. If it's in a particular venue you might have to get venue owners or TRF officials involved.
Yes you can have birthday parties call ahead and let them know and they also let you do weddings.
Definitely. Just make sure you call to arrange it ahead of time
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