After half an hour played, the visiting "Viola" had gone into a 2-0 lead and both goals were on penalties, the Chilean signed Erick Pulgar.
The first came when the nationally-qualified Swedish-Italian Gagliolo dropped Lorenzo Venuti and the second came when the ball took on Parma-back Matteo Darmian's hand.
Juraj Kucka reduced from eleven meters four minutes into the second half, but closer than that, Parma did not come even though Kucka fell with two minutes left and wanted another penalty.
Both Swedes were replaced in the match - Gagliolo after 74 minutes and Kulusevski eight minutes later.
Before today's game, Kulusevski had scored in two of the last three feuds.
The loss was Parma's fourth straight at home.