You are mistaken, you can enter and exit the Howard Beach station to the long-term parking lot from the East end of the station no charge or turnstiles. The west end has two sets of turnstiles for exit to the street and one for the subway (A train).
Check Google Earth and you can see the crosswalk in the long-term parking lot to the station at the east end (direction of train travel towards the terminals) and I've parked near that station in bad weather since there is a covered walkway (lined up with that crosswalk). It is also next to designated handicap parking spaces.