Parssinen also had an assist for the Predators (42-31-8), who missed the playoffs. Juuse Saros made 21 saves.
"I think, with this particular group over the last six weeks, it's just been a group that's, they've been hungry. They've been tenacious. And, you know, they play with a lot of pride," Nashville coach John Hynes said. "And I thought we did those things tonight. It was good to see."
Mark Jankowski made it 1-0 at 8:02 of the first period with a short-handed goal, tapping in a pass from Tyson Barrie who split the defense on the rush.
Jon Merrill found Gaudreau for the one-timer to tie the game 1-1 at 14:19.
Kiefer Sherwood puts Nashville back ahead 2-1 at 16:19, potting the rebound after Tommy Novak beat Merrill in close with his shot stopped by Gustavsson.
"We know that they play man on man so I think if we cut back and burn a guy, then you should have an opportunity," Sherwood said. "I got a fortunate bounce."
Gaudreau tied it 2-2, slapping the loose puck past Jaros after he made the initial stop on Marcus Foligno at 10:46 of the second period.
Yakov Trenin made it 3-2 after capitalizing on a bad line change for the Wild in the neutral zone at 14:04 of the third period.
"Their d-man made an awkward play, I thought he just went for the puck but he went for a change, I couldn't believe my eyes," said Trenin. "And then we get as many passes as we can just to shake the goalie and good things happen."