Ilya Kovalchuk scored a short-handed goal in the second period and added an assist on Parise's winner for the Devils (24-17-2), who jumped over Toronto into sixth place in the Eastern Conference.
Ales Hemsky scored for the Oilers (16-22-4), who are falling deeper out of contention in the West. Edmonton has lost four straight and 11 of 13.
New Jersey took a 1-0 lead eight minutes into the second when Kovalchuk picked off Jeff Petry's pass and went in on a short-handed breakaway. Kovalchuk beat Khabibulin high to the glove side for his 17th goal of the season, and New Jersey's NHL-leading 11th short-handed tally.
The Devils nearly doubled their lead with three minutes remaining in the second when Elias tipped a shot through Khabibulin's legs, but the goalie reached behind in time to smother the puck.
Edmonton tied it 8:32 into the third when Hemsky fired a shot from the top of the circle on the power play and beat Hedberg for his fourth of the season.
NOTES: It was just the third time the Devils played in Edmonton since Dec. 30, 2001, and was the only meeting between the teams this season. ... Oilers leading scorer Jordan Eberle remained out with a knee injury, but he might start skating soon and return by the All-Star break at the end of January. ... The Oilers brought up forward Milan Kytnar from the Stockton Thunder of the ECHL after Gilbert Brule was claimed on re-entry waivers by the Phoenix Coyotes on Tuesday. Ryan O'Marra sat out because of the flu, and Kytnar made his NHL debut. ... The Oilers were also without defensemen Ryan Whitney (ankle), Tom Gilbert (knee) and Cam Barker (ankle). ... New Jersey was missing C Travis Zajac (Achilles) and D Andy Greene (foot).