Parenteau and Frans Nielsen beat Jets goaltender Ondrej Pavelec in the shootout, while Islanders goalie Evgeni Nabokov stopped Blake Wheeler and Kyle Wellwood on their attempts.
Parenteau and Michael Grabner gave the Islanders a pair of one-goal leads in the first two periods, but Nik Antropov and Andrew Ladd responded for the Jets.
Pavelec kept the Jets (15-13-5) in the game with acrobatic saves from start to finish. He stopped 31 of the 33 shots he faced.
Nabokov stopped all 19 shots he faced for the Islanders (11-14-6) in relief of starting goaltender Al Montoya.
Montoya made 17 saves on 19 shots before a collision with Evander Kane late in the second period forced him to leave the game. Kane was given a penalty for goaltender interference despite arguing that defenceman Milan Jurcina pushed him into Montoya.
The Jets would have moved into a tie with Ottawa and Toronto for seventh place in the log-jammed Eastern Conference with a win, but instead are in a ninth-place tie with Washington and Buffalo.
The Islanders had the game's only two power plays — including one that came with four minutes remaining — but couldn't score on either attempt.
New York dominated the game early, generating a handful of good scoring chances before Parenteau one-timed a Matt Moulson pass into a wide-open net to make it 1-0.
The sluggish Jets tied it with two minutes left in the first period when Antropov redirected a Johnny Oduya point shot past Montoya.
Grabner, a former member of the Manitoba Moose, was in the right place at the right time when he jumped on a loose puck in the crease to give the Islanders a 2-1 lead in the second period.
Ladd had the Jets' reply near the midpoint of the period with a long wrist shot that beat Montoya high to the glove side.
The Jets finish their six-game home stand with games against Montreal on Thursday and Pittsburgh on Friday. The Islanders face the Rangers at Madison Square Garden on Thursday.
Notes: Forward Antti Miettinen played his first game for the Jets since he was claimed off waivers from Tampa Bay on Dec. 13… Jets centre Bryan Little missed his first game of the season after taking a shot off the foot on Saturday… Forward Brian Rolston and defenceman Steve Staios were both out of the Islanders lineup after suffering concussions in the last week.
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