Brian Rolston and Parise scored 3:29 into the game, Brian Gionta made it 3-0 midway through the first period, and Scott Clemmensen finished with 25 saves as New Jersey won for the second time in five games.
Alex Burrows also scored for the Canucks, who have one win in their past six games and have dropped five straight on home ice.
Despite being called out by coach Alain Vigneault for their recent lack of urgency and desperation, the Canucks were sloppy defensively and behind early.
Shortly after ringing a shot off the post, Rolston scored easily on a one-timer from the slot 1:05 into the game as goalie Jason LaBarbera scrambled awkwardly back into position after handling the puck behind the net.
Parise put a rebound into an empty net 2:24 later after LaBarbera got a piece of Zajac's blocker-side shot off a 2-on-1 rush, ending the Canucks goalie's night after just four saves.
Replacement Curtis Sanford didn't fare much better, turning the puck over behind his goal and leaving Gionta with an empty net 11:07 into the game.
Booed off the ice by a sellout crowd of 18,630 after the first period, Burrows solved Clemmensen on a rebound midway through the second. But Zajac got that back four minutes into the third period, converting a 3-on-2 shortly after Canucks defenseman Willie Mitchell hit the post at the other end.
Demitra, who started the game demoted to the fourth line, scored with 7:41 left on a rebound of a Mats Sundin shot. It was the second point in four games for Sundin, who scored his first as a Canuck in Saturday's 4-2 loss to San Jose.
Clemmensen, getting his first chance as a No. 1 goalie at the age of 31, improved to 16-9-1 as the Devils continued to win without star goaltender Martin Brodeur.
New Jersey improved to 19-13-1 since the three-time Vezina Trophy finalist tore his biceps on Nov. 1.
Notes: Vancouver LW Taylor Pyatt returned after missing eight games with a broken foot. Canucks C Rick Rypien has recovered from sports hernia surgery, but coach Alain Vigneault said after the morning skate he has been given an indefinite leave of absence to deal with an undisclosed personal issue. New Jersey is 2-1-0 midway through a season-long six-game road trip that continues in Columbus on Friday.