Is overnight parking allowed? Is this the closest parking for false Cape state park?

I don't think overnight is allowed as the whole park has closing hours.

If you are camping in False Cape, you get a permit and leave one in your car so it can stay parked here.

The closest parking for False Cape is Back Bay Wildlife refuge. They lock the lots at little island.

They close the entrance and exit after it's seen the last car is gone. Then it's locked. Hope this helps answer your question.