Andrea Bertolacci struck the post twice in the second half as Genoa were denied what would have been an important win in the battle against the Serie A drop.
A goalless draw suggests a game of little attacking invention but there was no shortage of chances in an open match.
Bertolacci went closer than most when he hit the woodwork in the 48th and 57th minutes, but neither side could find the net.
Bertolacci had the first meaningful chance of the day, meeting Eros Pisano's cross at the near post and drawing a diving save from Nicola Pavarini.
Parma returned fire through Amauri, but his shot lacked the necessary power.
The second instalment in what was to be an afternoon of frustration for Bertolacci came in the 27th minute when he was agonisingly close to putting the final touch on Marco Boriello's cross.
Sebastien Frey kept the hosts at bay 10 minutes later when he got fingertips to Ishak Belfodil's headed effort, before Boriello sent the final chance of the half over the bar.
The second half saw both sides start busily, Sotiris Ninis wasting an early opening for Parma before Bertolacci lined up a 25-yard strike which beat Pavarini but not the far post.
Bertolacci looked comfortably the man most likely to break the deadlock and nine minutes after he was first denied by the frame of the goal he again clattered the post, this time with a right-footed effort from the centre of the area.
Genoa would have been good value for a goal but could not find the finish - Marco Rigoni and Borriello both spurning decent chances before the hour mark.
Parma were not making up the numbers, though, and Frey produced a fine stop to keep out Amauri's close-range attempt.
Indeed, the hosts ended the game in the ascendance, Jaime Valdes and Jonathan Biabinay each going close to a winner.