With just one win in the previous nine games, New York entered with a tenuous three-point lead over Montreal for the final Eastern Conference postseason spot.
But rookie Justin Mapletoft started New York's comeback on the next shift and Jason Blake forged a 2-2 tie 6:32 into the second period.
Shawn Bates, who had a goal waved off in Tuesday's loss at Vancouver, netted his 13th off a turnover by former Islander defenseman Eric Brewer and Parrish scored his first since Feb. 11. It gave New York its first two-goal cushion in two weeks.
Blake capped the scoring with an empty-netter as the Islanders remained three points ahead of ninth-place Montreal and moved five clear of the New York Rangers.
Garth Snow, who made his 18th start in 23 games, overcame a shaky first period and made 22 saves.