Orchard Beach is part of the NYC City Parks system, which has different rules from the City, which generally allows tripod use without a permit on outdoor city property by a group of fewer than five people, as long as it doesn't involve setting up lights, wiring, sets, etc., and doesn't obstruct roads and sidewalks. The City Parks guidelines are here--
http://www.nycgovparks.org/sub_newsroom/film_photo_guidelines/parks_film_photo_guidelines.html
Much seems to be left open to interpretation, if you don't have a permit, but usually when I've been questioned while photographing with a tripod on City Parks property, it's been about commercial use, and the officers have taken me at my word when I've said I was not shooting commercially.
Central Park seems most liberal about enforcement of these rules, allowing even commercial wedding photography, small-scale model shoots, and bird photographers with tripods and big glass, as long as it isn't too obtrusive and there aren't, say, officious production assistants with walkie-talkies keeping passersby out of the shot.
There are also NY State Parks within city limits, and they have different rules, and MTA property has its own rules.