Toronto kept the Wild at bay during nine minutes of power-play in the third period, forcing as many shots as Minnesota.
But with 4:17 remaining in the third the Wild tied the game, Charlie Coyle setting up Parise for his ninth of the season.
Maple Leafs goaltender Jonathan Bernier finished with 33 saves but failed to get a fourth career shutout against the Wild despite making six saves in the first period, 14 in the second and eight in the third before Parise finally beat him to tie up Mason Raymond's score in the second.
The Pittsburgh Penguins suffered a third straight loss with a 2-1 defeat to the Philadelphia Flyers at the Consol Energy Center.
The Penguins still lead the Metropolitan Division by one point from the Washington Capitals and seven ahead of the Flyers who continue to climb away from the basement.
Brayden Schenn scored twice for the Flyers who had a third consecutive win, Pittsburgh suffering a third straight loss despite a goal from Sidney Crosby who tied it up in the second before Schenn got his second.
Ray Emery stopped 30 of the 31 shots he faced in his second win of the season for the Flyers.
The Dallas Stars beat the Edmonton Oilers 3-0 which was a third straight shutout for the home side at Rexall Place.
Rich Peverley, Tyler Seguin and Antoine Roussel scored for the Stars and Kari Lehtonen made 22 saves to collect his first shutout of the season and 23rd of his career.