The spotlight wasn't on Parise that night, though.
It was on another rookie forward who was playing in his first NHL game: the No. 1 pick in the 2005 NHL Draft, Sidney Crosby of the Pittsburgh Penguins.
"It was my first game and his first game," Parise said. "All eyes were on him, not on me."
There likely will be just as much fanfare Friday with Parise expected to play in his 1,000th NHL game when the Minnesota Wild visit the Dallas Stars at American Airlines Center (8:30 p.m. ET; FS-SW+, FS-N, FS-WI, NHL.TV).
That's fine with Parise.
Trailing the Calgary Flames by four points for the second wild card into the Stanley Cup Playoffs from the Western Conference, the Wild are trying to climb back into the postseason picture, and that's Parise's primary focus.
"We're going to have our hands full the next little bit," Parise said. "It will be a fun little stretch, though. We've got to start winning some games."
That doesn't mean Parise won't enjoy reaching 1,000 games. The 35-year-old has been through a lot during his 15 NHL seasons, including injuries to his right knee and back, and understands the significance of playing that many games.
"As you get older and play longer, I think you appreciate it more because you see it's not easy to get there," Parise said. "Not everyone gets there. You need a lot of things to go your way. So I appreciate the number a lot more than I did before."
Through his first 999 games with the Devils and the Wild, Parise has 779 points (381 goals, 398 assists). A six-time 30-goal scorer, he leads Minnesota with 20 this season and is third on the Wild with 33 points in 53 games.
At 5-foot-11, 195 pounds, the Minneapolis native has never been afraid to go to tough areas in the corners and in front of the net to make plays and score goals, and his work ethic continues be a driving force for the Wild.
"He's the same size as me, but he makes his living in front of the net," Minnesota goalie Alex Stalock said. "You're not going to do that unless you're competitive. A lot of his goals in the last 1,000 games, the goals he's scored, I don't know how many he's finished on his feet. Whether he's on his stomach, on his back, on his knees, whatever it is, he's sticking his nose in there and competing for the last whack at the puck."
Video: VAN@MIN: Parise buries Suter rebound in front for PPG
Parise's journey to 1,000 games began with the Devils, who selected him with the No. 17 pick in the 2003 NHL Draft. Facing Crosby and getting the winning goal and an assist in a 5-1 victory in his debut were fun, but what Parise remembers most from that game was playing against Hockey Hall of Famer Mario Lemieux.
"My 'This is the NHL' moment was in warmups," he said. "We were doing breakouts and Pittsburgh was doing breakouts too and I look back and I see Mario. Playing against Mario in my first game, to me, that was the guy I watched growing up and one of the best all-time. I watched him and now I'm on the ice against him for the first game."
Parise points to the Devils' trip to the Stanley Cup Final in 2012, when he was captain and they lost the Los Angeles Kings in six games, as the on-ice highlight of his seven seasons with them.
"But I think it was just the friendships," Parise said. "You're pretty lucky to have a lot of really good teammates out there, some of the friendships made out there. I had a lot of fun playing and living out there."
Parise became an unrestricted free agent after that that season and he and his good friend, defenseman Ryan Suter, signed matching 13-year contracts with the Wild on July 4, 2012. Being able to play near his family in Minnesota was one of the main reasons for Parise's decision.
He and wife, Alisha, have had three children since moving there - 6-year-old twins Jaxson and Amelia and son Theodore Jean-Paul, who turns 2 on Saturday. Theodore Jean-Paul's middle name comes from his father, former NHL forward J.P. Parise, who died in 2015 of lung cancer.
Parise's father wasn't ill when he signed with the Wild, but as it turned out, the decision to sign with Minnesota allowed the younger Parise to spend more time with his father before he died.
"For him to get to spend time with my kids, his grandkids, for at least a year was really special," Parise said. "You don't think about those things at the time you're signing ever, but it worked out that way. I got to see him a lot more than I would have. I wouldn't have been able to see him. I would have been flying back and forth. Just that thing alone made coming here worth it."
But ultimately, Parise wants to win the Stanley Cup. The playoff highpoints for him with the Wild so far were upsets of the Colorado Avalanche in the Western Conference First Round in 2014 and the St. Louis Blues in the first round in 2015.
Minnesota hasn't advanced past the first round since, and is in danger of failing to qualify for the playoffs for the second straight season. Parise hasn't given up on getting back to a Cup Final and winning this time, but knows the clock is ticking.
"Where I'm at now with five years left [on his contract], you crave that chance to get back there again, and it's hard," he said. "I'm not saying time is running out, but all the sudden you know when the end is coming. So that's what it's all about now, and always has been."
NHL.com independent correspondent Jessi Pierce contributed to this report