Since the merger of the five boroughs in 1898, New York City has many of the privileges of a State (granted by the NYS Legislature), one of which is the power to write its own gun laws. BB guns are illegal to possess in NYC, so there are no background checks.
Leaving aside the idiotic nature of NYC gun laws, shooting a BB gun in a public park seems to me like a very very bad idea.
I grew up in NYC, and these parks are typically small and full of small children running around. I started all 7 of my children with a CO2 rifle, and I enforced range discipline and safe handling rules from the beginning. I still don't allow CO2 rifle fire when there are kids playing in my yard (and I live in the woods in NH).
I don't think we know all of this story.
Plastic pellet airsoft is about as dangerous as a sucker dart gun. Probably less so...