Mauri, who has previously represented Italy at Under-17 level, is quoted by Goal.com as saying: "I can say that a dream has come true.
"Milan are a side that I have followed a lot because of the small rivalry that exists with Boca Juniors. My Milan idol was [Gennaro] Gattuso.
"The moment Milan are going through pushed me to come here. I need to improve, like Milan, and let's hope we can achieve this together."
Mauri scored two goals in 35 Serie A appearances for Parma last season, including the winning strike against Juventus in April.