The Rossoneri boss breathed a sigh of relief after the Frenchman scored twice to give his side their first win of 2015
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.
Any hopes of second equaliser were dashed late on as Menez teed up Cristian Zaccardo to score a third.
Parma entered the match 10 points adrift at the foot of the table and despite a promising performance slipped to a fourth straight defeat, intensifying the scrutiny on coach Roberto Donadoni as their dismal season - which has featured numerous issues on and off the pitch - continues.
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.
After much confusion, and conversation among the officials, Menez stepped up to send Antonio Mirante the wrong way from the spot.
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.
Michael Essien was introduced at the break and when his loose pass was greeted with boos by the home crowd it appeared the match may turn against the hosts.
But with 57 minutes on the clock, Menez was once again the man to step forward for Milan.
The Frenchman linked up wonderfully with Alessio Cerci down the right before cutting in and finding the bottom corner with a crisp left-foot shot.
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.
Menez received further reward for his tireless display in the 76th minute as he cut in from the right once again, but this time played the perfect pass for an on-rushing Zaccardo to crash into the net from eight yards and seal victory.