Elias scored the winner on New Jersey's final attempt. Parise beat Sergei Bobrovsky on the stick side on the first attempt.
Danny Briere scored the only shootout goal for Philadelphia. His goal was reviewed after he appeared to skid to a stop in front of Johan Hedberg, using the sweet fake to pound in the easy score.
David Clarkson tied the score at 3-all late in the third period after a costly turnover by the Flyers. Parise and Adam Henrique scored in the second to rally the Devils from a 2-0 hole.
Zac Rinaldo and Claude Giroux scored second-period goals for the Flyers. Max Talbot scored in the third on a penalty shot for a 3-2 lead.
Rinaldo and Henrique each scored their first NHL goals.
The Flyers played without key forward James van Riemsdyk after he suffered a lower-body injury a night earlier against Buffalo. Van Riemsdyk suffered a punishing hit behind the net on Wednesday and was slow getting up, the likely cause of the injury.
Danny Briere returned after missing two games because of an upper-body injury.
The Devils outplayed the Flyers in regulation that scored the clutch goals in the shootout. The Flyers picked Wayne Simmonds to take the first shot, skipping Jaromir Jagr. Giroux failed to score on Philadelphia's third attempt.
It came down to the shootout because the Flyers failed to make smart defensive plays.
Matt Carle's errant clear at the blue line was intercepted by Elias, who needed one pass to find Clarkson in front of the net to whip a backhander for the tying goal with 6:35 left in the third. It was Philadelphia's second costly turnover of the game that led to a goal.
Talbot had put the Flyers ahead 2:37 into the third after he was dragged down on a breakaway by Devils defenseman Mark Fayne. Talbot, 0 for 1 in his career on penalty shots, skated along the right boards, cut through the slot, and deked Hedberg with a forehand for the 3-2 lead.
Bobrovsky got the start in net a night after Ilya Bryzgalov held off the Sabres. Bobrovsky was sensational early - twice stopping Dainius Zubrus in the final 2 minutes of the first to keep the game scoreless.
With defenseman Chris Pronger still out with an eye injury, Bobrovsky was battered with 30 shots though two periods. Hedberg only faced 14.
Rinaldo and Giroux struck first and scored the game's first two goals. Rinaldo fired a wrister from the edge of the faceoff circle for his first career goal.
"Scoring my first goal is very special. It definitely gives me a little more confidence with the puck," Rinaldo said. "I don't just like hitting, I also like putting the puck into the net sometimes."
Jagr swept around the net and fed a perfect cross-ice pass to Giroux who connected for his eighth goal of the season. Hedberg got the start to give a rusty Martin Brodeur the night off after he suffered a 5-3 loss to Toronto.
Staked to a two-goal lead against the slumping Devils, Bobrovsky stumbled midway through the second period.
Parise backhanded a loose puck after a scramble near the faceoff circle for his fifth goal of the season that made it 2-1. Henrique capitalized off a Flyers turnover to match Philly's Rinaldo in memorable goals - and knot the score at 2-all.
The Devils took 21 shots in the second period - two fewer than Philadelphia's total through overtime.
NOTES: Zebrus and Hedberg are both former Flyers' draft picks. ... The Flyers scored their first penalty shot since Darroll Powe on Feb. 15, 2011. They scored their first home penalty shot since Simon Gagne on Feb. 14, 2009. ... The Flyers are 18 of 42 on regular-season penalty shots.