Watch: Parma draw stalls Milan's Champions League push
Samu Castillejo nodded Milan ahead after 69 minutes but Alves pulled the sides level with a curling free-kick three minutes from time.
Milan remain fourth on 56 points but Roma are two points behind in fifth before they play third-placed Inter Milan at the San Siro later on Saturday.
Atalanta, sixth, can pull level with Milan on points when they travel to second-placed Napoli on Monday.
Lazio are also lurking four points behind ahead of their game against relegated Chievo on Saturday with Torino seven points adrift of the Champions League places before travelling to Genoa.
Castillejo came off the bench after 66 minutes and had an immediate impact when he headed in a Suso cross three minutes later.
Champions Juventus can seal an eighth consecutive Serie A title later on Saturday, with the Turin giants needing just a point against Fiorentina for their 35th Scudetto.