The first chance of the game fell to the home side, when Andrea Bertolacci sent Ciro Immobile through, but he failed to find the target.
Moments later Immobile had another chance deflected over the crossbar by Parma goalkeeper Antonio Mirante.
Sebastien Frey then saved Parma defender Cristian Zaccardo's header from a corner.
The visitors eventually went ahead in the 27th minute when Alessandro Lucarelli scored with a left-footed volley.
Genoa were given a wonderful opportunity to pull level in the 68th minute when Jamie Valdes handled the ball inside the Parma penalty box.
However, Marco Borriello failed to score from the spot, first hitting the post and then seeing his follow-up shot smothered by Mirante.
The Italian striker, though, did eventually make amends for his earlier miss when he stepped up to take another spot kick on 88 minutes.
Lucarelli brought down Borriello inside the area, and the Genoa striker made no mistake from 12 yards, earning his side a 1-1 draw.