The France international added to his strike on the opening weekend of the season with a first-half penalty and an audacious back heel late in the game to seal victory.
Having touched the ball past Antonio Mirante the forward chased along the goalline before executing an outlandish effort that flummoxed the Parma goalkeeper.
Menez moved to Italy after a frustrating season with Paris Saint-Germain last season and says he is loving life under Filippo Inzaghi.
Playing in a central attacking role in place of the injured Fernando Torres the 27-year-old lit up the Stadio Ennio Tardini with some sparkling attacking play but acknowledged tha Milan must do better defensively.
"We played a good game and ran a few too many risks, as we had the initiative throughout and the four goals we conceded were a little strange," he told Sky Sport Italia. "In the end it's the win that counts.
"The team showed courage and never let go. I am trying to do all I can and help the team. Tonight we suffered a lot too and must work on that.
"I feel good, this team and the coach helped me a great deal and I am happy at Milan."
Milan's win sees them keep the early pace with Juventus and Roma, with the defending champions next up for Inzaghi's men at San Siro.