The shutout was Brodeur's third of the season and the 83rd of his career, moving him into sole possession of fourth place on the NHL's all-time list, one ahead of Hall of Famer Jacques Plante.
The Devils' shutout streak follows a three-game losing streak in which they gave up 15 goals.
Columbus, which was playing its second game in as many nights, only had a few good scoring chances. Rick Nash had the best chance from between the circles on a first-period power play, while Nikolai Zherdev was stopped in close on a rebound on the first of two power plays in the final five minutes. Nash also blistered a short-handed shot from the right circle with less than a minute to go that hit Brodeur in the chest.
Parise broke up a goaltending duel between Brodeur and Pascal Leclaire with his second goal in as many games and his fifth of the season. It may have gone into the net off a Columbus stick.
With the Blue Jackets unable to get the puck out of their own end, Parise exchanged passes with defenceman Brian Rafalski before finding Travis Zajac between the circles. The rookie's shot missed the net and bounced off the end boards to Parise at the right side of the net.
With Leclaire scrambling to get back to his feet, Parise's stuff shot caromed into the net off the stick of either Columbus defenceman Duvie Westcott or Zherdev.
Until then, Leclaire was outstanding in his bid for a second straight shutout. He stopped 38 Kings' shots on Friday for his first career shutout. He finished with 20 saves as the Blue Jackets lost for the fifth time in six games.
Leclaire stopped Patrik Elias in close in the opening minutes and Sergei Brylin early in the second period on a power-play setup by Elias.
Notes: Devils C Scott Gomez was sidelined with groin injury suffered Thursday against Florida. Brylin replaced him on the top line with Brian Gionta and Elias. ...This was the only game between the teams this season, and Columbus' first trip to the Continental Airlines Arena since, Oct. 12, 2002.