If you are getting one of the pavillions, I believe you need t reserve it and pay for it, but if is a small group I think you can just get one or 2 picknick tables from anywhere in the park. For free. By the big pavillion, by the train station, the zoo entrance, in front of the Japonese Tea Garden, baseball fields, by the river or by the trails. ItI think if you get there by 5pm you get still get to choose. You can also bring your own table and set it whatever you please. I love this park. I walk there every morning. Bring bug spray, just in case.
Call Parks and Rec and set up a reservation to guarantee your date.
Plenty of space by the train depot, although homeless usually running amok through the area naked and angry...
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