For us, the northern part of Central Park (between 100th and 110th) is a favorite, specifically the big circle of grass at the top of the Great Hill (enter at 106th and walk up the stairs). There are picnic tables on the western edge of the circular path and grassy spots all around the main loop that are (usually) sparsely populated.
We've also been to a party under a huge willow tree next to Turtle Pond at the bottom of the Great Lawn, near Belvedere Castle. It was pretty easy to find and much less crowded than the Sheep's Meadow.
If there are hidden gems in the big parks (Prospect, Central, Riverside, Flushing Meadows) that you've found, let us know. But maybe your favorite picnic spot is a quiet patch of grass off a main road? Or a playground that has huge tables? Any public grills that are especially nice?
We're always looking for new places to hang out, so give us your favorites.