The New York City Parks Department manages all NYC Parks properties. They are a part of the City administration and funded with taxpayer money.
Parks are protected in New York State for the use of people as open space. A City in the State cannot decide to simply close down a park and use the land for something else without getting approval from the state government after land has been dedicated as parkland.
In order to convey parkland to a nonpublic entity, or to use parkland for another purpose, a municipality must receive prior authorization from the State in the form of legislation enacted by the New York State Legislature and approved by the Governor. “Once land has been dedicated to use as a park, it cannot be diverted for uses other than recreation, in whole or in part, temporarily or permanently, even for another public purpose, without legislative approval.” United States v. City of New York, 96 F.Supp.2d 195, 202 (E.D.N.Y. 2000). This clear law has been applied consistently since 1871.
You can see the State's handbook on alienation here.