This match was decided in the final minute of stoppage time by substitute Sergio Pellissier.
Chievo had the first shot on goal, although it was almost accidental, when Emanuele Pesaresi's mis-placed cross was heading under the crossbar but Sebastien Frey managed to tip it over.
Parma had their penalty appeals waved away when Antonio Benarrivo's effort struck Nicola Legrottaglie's arm.
The Gialloblu almost took advantage of an error, but goalkeeper Cristiano Lupatelli recovered from his poor first attempt to block an Emiliano Bonazzoli header.
Just before the break Chievo hit back, but Oliver Bierhoff's volley shaved the crossbar from a good position.
It was a very even match and at the dawn of the second period another Lupatelli error handed the ball to Filippini inside the box, but a defender's body blocked the shot.
Adrian Mutu has been on fine form recently and saw another close-range shot parried by desperate defending.
But Parma have often conceded late goals this season and fell into the same trap.
In the final minute of stoppage time substitute Pellissier latched onto Franceschini's low cross, though Matteo Ferrari was a little slow to catch up to the Chievo man.