Evgeni Nabokov stopped 37 shots in the win, which gave the Islanders two victories in their last three tries.
Matt Martin and Matt Moulson also scored for the Islanders, who won despite being outshot 38-25.
Chris Thorburn opened the scoring late in the first period with the Jets' only goal of the game.
Moulson tied it with a power-play goal at 3:55 of the second period and Parenteau put the Islanders ahead for good at 1:36 of the third period.
Parenteau then assisted on Martin's empty-netter to close out the scoring.
Notable deaths of 2014 [PHOTOS]