Palacio and a strike from Fredy Guarin had put Inter 3-2 ahead in the second half but their lead lasted just four minutes, with a Sansone brace and a goal from Marco Parolo capping a compelling six-goal encounter.
The visitors started the brighter of the two sides and took a deserved lead after just 11 minutes, when Sansone broke the offside to take advantage of Marco Marchionni’s well-time through ball and blast past Samir Handanovic in the Inter goal.
Palacio passed up a glorious chance to get his side back on level terms, latching on to Guarin’s dragged shot to the back post. But he scooped his shot on goal over the bar with the net gaping.
However, Inter grew into the contest as the first half progressed and finally found an equalizer when Palacio got on the end of Jonathan’s driven cross into the six yard box.
But in stoppage time at the end of the first half Parma restored their lead when Handanovic fumbled a tame cross straight at the feet of Parolo, who had the simple task of tapping into the empty net.
Inter equalized through Palacio after 54 minutes, when he jumped highest to meet Ricky Alvarez’s dangerous cross into the box and power the ball past Antonio Mirante.
And just a minute later Walter Mazzarri’s side completed a remarkable turnaround, making it 3-2 when Guarin’s shot was deflected into the net by the helpless Alessandro Lucarelli.
However, Parma struck back almost immediately, with Sansone emphatically finishing off Mattia Cassani’s low pass into the box.
Mazzarri replaced Javier Zanetti with Yuto Nagatomo in an attempt to give his side more of an attacking outlet down the right side, but it was Parma who came closest to finding a winner, with Handanovic tipping Afriyie Acquah’s dipping shot over the bar.
Andrea Ranocchia made a goal-saving block with six minutes remaining, deflecting Jonathan Biabiany’s shot from inside the box past his own post with the Inter defence absent.
However, the final chance fell to Esteban Cambiasso, who was afforded a free shot at goal from eight yards out in stoppage time. But he smashed his volleyed effort straight at the Parma keeper Mirante.