Martin Brodeur blanked Philadelphia's Simon Gagne and Petr Nedved in the tiebreaker.
The Devils tied it at three with 2:12 left in the third period when Brad Lukowich's short-handed wrist shot sailed by goaltender Antero Niittymaki, who was screened and knocked to the ice by New Jersey's Jay Pandolfo.
Mike Knuble had scored the go-ahead goal for the Flyers earlier in the period.
Todd Fedoruk celebrated his return to Philadelphia with a goal and an assist, and Randy Jones also had a goal and assist for the Flyers, who had won three of four.
Parise and Patrik Elias scored the other goals for the Devils, who snapped a season-high, four-game winless streak (0-3-1) with a 5-2 home win over Pittsburgh on Friday.