Mauri joined Parma from Ferro Carril Oeste in 2011 and became a key figure for the Ennio Tardini side last term.
The youngster scored twice in 35 Serie A appearances for Parma in 2014-15, with his strike in the 1-0 win over Juventus back in April the highlight of his season and is now looking forward to the next chapter of his career.
"I can say that a dream has come true," Mauri told reporters. "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."