The hosts had a chance after just two minutes when Massimo Gobbi found some space in the area, but the defender could only put his strike over the bar.
Genoa took the lead against the run of play on 22 minutes when Isaac Cofie found space in the box and fired into the bottom left corner.
The away side continued to push forward and troubled the Parma defence with crosses from Paolo De Ceglie.
The home side equalised just after the half-hour mark when a rebound in the area fell kindly to Ezequiel Schelotto, who finished well.
Marco Parolo almost put Parma ahead on 38 minutes with a powerful strike from distance, which whistled past the post.
Five minutes into the second half, Gianni Munari created a good opportunity for Parma with a snap shot from the edge of the area, but his effort went just wide of the post.
Emanuele Calaio threatened for Genoa on 57 minutes when he rose above the defence to head towards goal, but his towering attempt bobbled wide.
Amauri wasted a fine chance on 66 minutes when he was found by an inch-perfect cross, but the striker just couldn't find the target with his header.
Parma pushed forward for a late winner and created another chance on 75 minutes when Parolo quickly adjusted himself to get a header on goal, but it went wide of the left post.
Perin kept Genoa in the game on 85 minutes by producing a good save to deny a header from Amauri.