I’m just guessing here, but don’t most state parks close at dusk?
Here’s the form:
https://parks.ny.gov/regions/attachments/2020LongIslandPermits.pdf
It’s really nothing more than a parking permit that allows you to park and be in the park after dusk in order to stargaze and probably to prevent other sorts of unwanted activities that may take place after dark.
Some parks are open all the time (like Bear Mountain, just south of West Point), while across the NY Thruway I believe Sterling Forest State Park doesn’t open until 8 am. The former doesn’t allow hunting any time, while the latter allows it in season.
Lyme Disease (combined with other changes) has put a real dent in the popularity of camping compared to when I was younger; the deer are everywhere, and their ticks come with them.