People park on 62 St all the time, you will have no problem parking as long it's not over night, enjoy your time with your family.
Im not certain on the laws. But people park there everyday. I doubt you would get a ticket. Its always possible though.
I've wondered the same thing. I always park right next to the curb on the hill. I've never gotten a ticket, but I'm not saying it doesn't happen.
There is a parking lot on the 72nd side of the park. Then just a short path to the play grounds.
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