Newsday, citing FAA records, says the agency cited both pilots for failure to obtain clearance from air traffic control before entering the taxiway.
The Tampa, Fla.-bound jet got stuck at MacArthur Airport in Ronkonkoma on Dec. 27, 2012. No one was injured.
Both men in the cockpit recorded zero on the breath test for alcohol. The FAA placed them in a safety program, which it described as a "safety partnership" between the agency and airlines.
The pilots couldn't be reached for comment. They continue to work for Southwest.
The pilots association didn't respond to a request for comment.