To do so, they'll have to knock off Suter's former team, the Nashville Predators, who begin their road schedule Tuesday night trying to shake off back-to-back shootout losses.
Minnesota (2-0-0) is enjoying its best start since beginning 4-0-0 in 2008-09, but it has dropped six of its last seven meetings with Nashville, including two shootout defeats last season.
After Suter inked a 13-year, $98 million contract with Minnesota in July - the same terms as Parise - Tuesday will mark the first time he's skated against the Predators.
Suter, who led Nashville in ice time last season and totaled 236 points in his seven campaigns with the club, has yet to register a point with the Wild, though he has blocked seven shots through two games. The Madison, Wis., native made the move to Minnesota in part to be closer to family.
Parise, who is from Minnesota, has made an instant impact for the Wild, picking up two assists in a 4-2 season-opening win against Colorado on Saturday before scoring his first goal in a 1-0 win over Dallas on Sunday.
"It felt good to get one and get one early in the season," said Parise, who scored 31 goals for New Jersey last season and eight more in the playoffs. "You don't want to have that dragging over you."
The Wild were also bolstered by backup goaltender Josh Harding's seventh career shutout as he made his first start since being diagnosed with multiple sclerosis last summer.
"I can't say enough and I can't thank them enough for how good the team has been from the top down," Harding said. "I'm not just talking about the players. I'm talking about management, I'm talking about everybody. ... This is a family and I can't say enough about each and every guy in here."
Niklas Backstrom may be back in net against Nashville (0-0-2), which followed a 3-2 shootout loss to Columbus on Saturday with a 4-3 defeat in the same fashion Monday against St. Louis. Things won't get any easier as the Predators begin their longest road trip of the season, a seven-game trek that includes five contests against playoff teams from last year.
"We haven't been beaten in a game," coach Barry Trotz said. "We've been beaten in, a little bit you could say, a skills competition. But at the same time, it's huge, especially the extra points, I mean, in the league right now. If you can get them in a shootout, let's take them."
Despite the loss of Suter, the Predators' power play looks solid so far, converting 3 of 7 chances. Suter had a team-high 25 power-play points as Nashville led the league last season converting 21.6 percent.
Trotz said center Paul Gaustad is questionable for Tuesday's game after he was scratched against St. Louis with an upper-body injury.
Minnesota's Dany Heatley, who had two goals in the season opener and assisted on Parise's goal Sunday, has 23 points in 17 career games against Nashville, including nine on the power play. He scored at least one goal in each of his four games against the Predators last season as the Wild went 1-1-2.