Briere had 13 goals and 12 assists in 69 games last season. He also had three goals as Montreal advanced to the Eastern Conference final.
Avalanche executive Joe Sakic says the 36-year-old Briere brings ''experience to our hockey club.'' Briere was a first-round pick of Phoenix in 1996 and has played in 916 regular-season games with Philadelphia, Buffalo, Phoenix and Montreal.
Parenteau missed 25 games last season after hurting both knees during the year. He finished with 14 goals and 19 assists.