The Canadiens also receive a fifth round pick in the 2015 draft.
Parenteau finished with the lowest totals of his NHL career the past season, with 14 goals and 19 assists in 55 games. He added one goal and two assists in seven playoff games, his first taste of post-season action in the league.
The Hull native hit career highs with the New York Islanders in 2011-12, with 49 assists and 67 points. He was originally drafted in the ninth round by Anaheim in 2001 after playing in the Quebec Major Junior Hockey League.
It was hoped Briere would finish his career with a flourish back home in Quebec after being bought out of a lengthy contract he signed for Philadelphia, but he struggled and was demoted to the fourth line.
Briere, who turns 37 in October, had 12 goals and 13 assists in 69 regular season games and added three goals and four assists in 16 playoff contests.
The Gatineau native finished the season 21st among active players in career goals, and his next NHL goal with be his 300th.
Parenteau has two years left on a deal that represents a salary cap hit of $4 million US, while Briere has one year left at the same rate.