Mergim Vojvoda's 63rd minute strike sealed the win and lifted Torino three points clear of the relegation zone. They also have a game in hand.
Parma's fifth successive defeat left them 12 points adrift of 17th-placed Cagliari, who occupy the last safe position in the league with four matches remaining. Cagliari also have the better head-to-head record which is a deciding factor in Serie A.
Parma, who have 20 points, follow Crotone in being relegated. One more relegation place is still to be decided.