Patrick Marleau scored with 1:41 left for San Jose, which went 1-3 on a four-game trip. The Sharks, who have lost three in a row in regulation for the first time this season, are 0-5-1 in their last six games in Minnesota.
Antti Niemi made 10 saves.
Parise and Koivu scored second-period goals to give Minnesota a 2-0 lead. Parise pushed the Wild's edge back to two goals when he scored at 19:54.