We are a week behind you.
We try to get to the parking lot 2 hours early. We have parked in the JFK Long Term Lot, which was fine - a quick train ride to terminal 4. Recently we have started using some of the independent guys. They can appear sketchy, but there are several who get good reviews on Yelp. Last time we parked with AirLot: $10 per day payable in cash up front, van ride directly to the terminal (faster/easier/warmer than the train), and a ride back to car with no waiting. C & D is another good one.