Parker, a three-year starter, led the Crimson Tide to a 12-0 regular season as a senior before closing his career with losses to Florida in the Southeastern Conference championship game and Utah in the Sugar Bowl.
He was among 10 players signing with the Falcons after being passed over in the draft. The others were San Diego State receiver Darren Mougey, Clemson receiver Aaron Kelly, Colorado defensive end Maurice Lucas, Auburn long snapper Robert Shiver, West Virginia offensive lineman Ryan Stanchek, Arkansas offensive lineman Jose Valdez, Florida State linebacker Derek Nicholson, Northern Arizona punter-kicker Robbie Dehaze and Missouri linebacker Brock Christopher.
In addition, the Falcons signed receiver Troy Bergeron, who spent three seasons with the Georgia Force in the Arena League.
The Falcons released defensive end Simon Fraser and defensive tackle Kindal Moorehead after bulking up those positions in the draft. Moorehead played in 14 games, making 18 tackles and one sack. Fraser played in six games.
The team waived defensive back Darius Vinnett and offensive lineman Nathan Bennett, who spent the last month of the season on the practice squad.