Andrea Gasbarroni put the hosts ahead from the penalty spot after Brazilian striker Reginaldo was clipped by Cristiano Zanetti inside the area.
Gasbarroni turned from scorer to provider in the 57th minute, setting-up Andrea Pisanu for the second.
Nicola Legrottaglie pulled one back 15 minutes from time when he headed home at the near post.
Domenico Morfeo was involved in a scuffle with Juve defender Giorgio Chiellini and both were sent off
Vincenzo Iaquinta, who replaced Alessandro Del Piero, grabbed the equalizer seven minutes later to salvage a point for Juventus.
Parma defender Ferdinand Coly saw red in the dying minutes for a tackle on Zanetti.
The result put Juventus in third place of the Serie A table with 22 points, three adrift of leaders Inter Milan, whose match with Lazio was called off after a Biancocelesti fan was shot dead by a policeman.