I might be mistaken, but I do not think that there is any way to get to or from the AirTrain platform at Howard Beach station without passing through the fare-gates. If you want to access the long-term parking area without paying the AirTrain fare, I believe that you must use the Lefferts Boulevard AirTrain station.

As a side note, if you wanted to travel between Howard Beach and Jamaica stations via AirTrain, it would cost you $15.50: $7.75 to enter at one station, and $7.75 to exit at the other.