No, it wasn't the actual first park, not by any means. That honor goes to Bowling Green. First Park didn't even get its name until a 1997 renovation. But it has the benefit of its location at the First street and the First avenue in the First city. (We New Yorkers do like to think of our city as the First – in importance.)
The New York City Parks Department website uses its entry on First Park as an opportunity to expound upon the history of playgrounds in the city. The phrases "Robert Moses" and "years of neglect and disrepair" figure in the description, as they seem to everywhere in NYC history. I'd like to take this opportunity to mention my appreciation for the anonymous scribes who create the Parks Department park descriptions, which are posted on the website and also, often, on signs in the parks.
In fact, First Park isn't all playground. The southern tip features this attractive expanse of asphalt where people can sit, far enough from the play areas that you know they're not here keeping an eye on their children. Passive enjoyment, then; there you go. To the left is a typical London plane tree with particolored bark.