Isaac Cofie gave the visitors a lead which lasted just 10 minutes before Ezequiel Schelotto levelled.
Parma searched more insistently for a winner while Genoa also had chances to take all three points, but there were no further goals as the two sides took a point apiece, Parma remaining behind Fiorentina and Inter in the search for a place in Europe next season
Parma started the game brightly with two good early chances to take the lead.
Gobbi wasted the first after exchanging passes with Amauri only to send his shot into the sky.
Just a few minutes later, Schelotto sent in a cross from the right which Amauri missed by a matter of inches, and Mattia Perin collected.
Genoa lost Francesco Matuzalem to injury after just 15 minutes, but they took the lead six minutes later through Cofie, who left Antonio Mirante no chance with a shot into the far corner.
An acrobatic effort from Amauri just missed the target as Parma searched for an equaliser, which they found within 10 minutes.
Antonio Cassano's free-kick was sent back into the danger area by Alessandro Lucarelli with the ball finding its way through to Schelotto, who sent his shot into the roof of the net.
Gianni Munari and Emanuele Calaio went close with both sides clearly keen on the points and not settling in any way for a draw.
Amauri and Marco Parolo wasted further chances for Parma, who almost fell behind in the 76th minute when Paolo De Ceglie's deflected cross landed on the top of the crossbar.
Parma launched a late rally, but Perin proved decisive with saves from Amauri and a deflected Raffaele Palladino effort denying the hosts victory.