Varlamov yielded a score-tying goal to Joffrey Lupul late in the second period, but then shut the door, stopping all nine shots he saw in the final period.
Parenteau broke a 1-1 tie with 2:30 remaining in the third, converting a goal-mouth pass from Jamie McGinn for the decisive tally.
Cory Sarich also scored for the Avalanche, who improved to a 3-0-0 under new Coach Patrick Roy, a franchise-best start to a season.
Jonathan Bernier turned aside 31-of-33 shots for the Maple Leafs, who dropped their first of the season after beginning 3-0-0.
2014: The Year in Music [PHOTOS]