Hernan Crespo gave Parma an early lead against one of his former clubs, but Inter was soon 3-1 ahead thanks to two goals from Stankovic and another from Esteban Cambiasso. Crespo grabbed his second before halftime, but in the second period Thiago Motta came off the bench to make it four for the home side before Stankovic completed his treble.
Parma hit the frame of the goal on two occasions and Inter goalkeeper Luca Castelazzi twice denied Crespo his hat trick goal.
"We came out slowly and conceded an early goal," Stankovic said. "But we reacted in the right way with our goals." Parma took the lead in the fourth minute. Angelo crossed from the right and Crespo raced in ahead of Lucio and Marco Materazzi to clip the ball over Castelazzi.
Inter equalized in the 18th. Stankovic's shot from the edge of the area deflected off Alessandro Lucarelli to leave Parma goalkeeper Antonio Mirante wrongfooted. Less than a minute later Stankovic doubled his tally.
Jonathan Biabiany raced into the Parma area and laid the ball off to the Serb, who hit a low shot into the corner of the net. Five minutes later Inter scored its third. Biabiany flicked on a corner by Wesley Sneijder and the ball fell to the unmarked Cambiasso, who poked it home.
Crespo brought Parma back into the match with his second in the 36th. Materazzi failed to clear, allowing Antonio Candreva to slip the ball to Crespo to fire low into the corner. The Argentine nearly grabbed a hat trick two minutes later. Again he was first to Angelo's cross, but Castelazzi pushed his header over the bar.
After the break the teams kept up the pace of the game and both sides had early opportunities to score. Cambiasso flashed a shot narrowly wide and Castelazzi again denied Crespo his hat trick when he was through on goal.
In the 52nd, Angelo hit the post and after Motta stretched Inter's lead with a scrambled finish, Crespo headed against the bar in the 74th.
"I'm delighted to be back and able to help the team," Motta said. "We needed a win to keep up with Juventus after its draw last night.
"We haven't seen the real Inter up to now, but now we are starting to see the real Inter as the injured players are starting to come back." Stankovic wrapped up his hat trick in clinical style.
Sneijder found him in space on the edge of the box and he slipped between Lucarelli and Cristian Zaccardo to power a shot past Mirante.