Inzaghi's men were facing a third straight home defeat but their top scorer stepped up in the key moments to secure a vital three points and lift them into the top half.
The Frenchman converted a first-half penalty before firing in a fantastic second 12 minutes after half-time to take his personal tally for the season to 12 goals.
This restored the hosts' lead after ex-Milan man Antonio Nocerino had cancelled out the opener, raising his hands in apparent apology to the sparse crowd inside San Siro in response.
New Milan signings Salvatore Bocchetti and Mattia Destro were in from the start as Inzaghi sought a fast start, although the visitors held them at arms' length for most of the opening stages.
That was until the hosts were awarded a 17th-minute penalty when Zaccardo's back-post header struck the arm of Cristian Rodriguez.
Rodriguez soon atoned somewhat for his part in Milan's opener, winning a free-kick in a dangerous position and meeting Daniele Galloppa's set-piece before Nocerino reacted first to poke in the knockdown from close range.
Milan responded well. Alex rattled the post with a powerful free-kick and Destro almost had his debut goal when Marco van Ginkel's cross caught out Mirante, though Diego Lopez was called into action by Silvestre Varela and Nocerino as Parma exploited the space left behind.
Milan were visibly lifted by the goal with Cerci exerting greater influence over the game - and Menez's work rate, as well as his goals, was beginning to win over some of the home fans.
