Both of Parise's goals came on bounces that barely made it over the goal line. Mikko Koivu and Matt Cullen each had a goal and assist for Minnesota, which won for the first time in Dallas since March 21, 2003, snapping a 16-game losing streak in Dallas (0-11-5).
With the game tied 4-4 in the third period, the Wild got a short-handed goal from Kyle Brodziak at 12:38.
Niklas Backstrom, who is 14-3-1 in his past 18 games and coming off a 2-0 shutout of San Jose on Saturday, stopped 22 shots for the win.
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