Parise scored the tying goal midway through the third period, then added another in the shootout to help Dubnyk and the Wild beat the Islanders 2-1 Tuesday night for their 10th straight win away from home.
Parise's shootout attempt hit the post to the right of Jaroslav Halak and trickled in after it bounced off the goalie's leg.
Dubnyk continued his stellar play with 37 saves and then denied all three shootout tries by the Islanders.
"We're not going to be perfect every game. You have to win the games when you're not," Parise said. "Devan was awesome tonight. He gave us every chance to win."
John Tavares broke a scoreless tie late in the second when he jammed his team-best 34th goal of the season past Dubnyk at 19:36. Kyle Okposo and Anders Lee assisted. The point was the 300th of Okposo's career.
But that was all the slumping Islanders could muster against Dubnyk. Tavares, Okposo and Frans Nielsen all failed to convert in the shootout.
Parise tied the game at 11:45 of the third with his 29th goal. Jason Pominville and Mikko Koivu had the assists.
The Islanders (44-25-5) had the better of play for most the game, outshouting the Wild 16-5 in the first period and finishing with a 38-32 advantage.
Still, the Wild improved to 42-25-7.
"We're a good hockey team. We built our game and were at our best in the third," Dubnyk said. "We're playing with confidence and it feels good to be contributing every night."
Dubnyk made his 32nd straight start for the Wild, an NHL-best 24-6-2 since Jan. 15 after the goalie was acquired from Arizona. And the Wild are 14-1-2 in their last 17 road games.
Halak was solid again after he shut out the Devils last Saturday to end a four-game losing streak for New York. He denied Pominville with a sliding stop midway through the second and stopped former Islander Thomas Vanek on a breakaway early in overtime.
"I thought we played a strong game," Islanders coach Jack Capuano said. "Guys played hard and games are tight this time of year."
Both teams had good chances in the extra session.
Tavares, who hadn't scored in five games, is trying to beat out Sidney Crosby and Alex Ovechkin for the NHL lead in points. Tavares also has 40 assists.
The game marked the start of a four-game homestand for the Islanders. They will face the Stanley Cup champion Kings on Thursday followed by back-to-back games against Anaheim and Detroit this weekend.
The Islanders are 23-12-1 at Nassau Coliseum this season. But since opening the season 17-4-0 at home, they are 6-8-1 in their last 15.
They have lost five of their last six, totaling only eight goals in those games.
"We just have to find a way to get goals," defenseman Johnny Boychuk said. "They started to play well in the third and they had a good break on their goal. We did some good things. We just have to bear down with our opportunities."
NOTES: New York is 25-7-5 in games decided by one goal. The Islanders are 7-4 shootouts. Halak was handed his first shootout loss after five wins. ... Before the game, the Islanders honored Hall of Fame center Pat Lafontaine, who scored 287 goals for the team from 1984-91. ... The Islanders scratched defensemen Lubomir Visnovsky and Matt Donovan plus forwards Colin McDonald and Michael Grabner. ... The Wild scratched forwards Ryan Carter and Kyle Brodziak, and defensemen Christian Folin and Nate Prosser.