Parise, stationed at the post to Dubielewicz's left, scored the winning goal during a 5-on-3 advantage with 3:44 remaining in the second. Andy Greene's point shot hit that post, and the puck bounced right to Parise for a tap-in goal.
The Islanders (2-4-2) scored first when Chris Simon, behind the net, slipped the puck to Kip Brennan in front. Veteran NHL defenseman Bryan Berard, on a tryout with the Islanders, also assisted.
New Jersey (4-2) answered on the power play with 1:34 remaining in the first. Parise went behind the net and fed the puck on his backhand to Zubrus at the right post, who hit the open side of the net.
The teams played in the home of the Bridgeport Sound Tigers, the Islanders' AHL affiliate. Seven players in New York's lineup, including Dubielewicz, were former Sound Tigers.