The deal with Parker, the 26th overall pick, was completed just before the deadline Monday night.
Colorado also agreed to terms with right-handed pitcher Peter Tago, who was taken at No. 47.
Parker displayed his two-sport versatility last season, throwing for 2,526 yards to help the Tigers win the ACC’s Atlantic Division and reach the conference title game.
In the spring, the strong-armed outfielder hit .358 and propelled the Tigers into the College World Series.
He’s the first Division I athlete to throw 20 touchdowns and hit 20 homers in the same academic year.
The 6-foot-1, 200-pound Parker went through football practice with the Tigers on Monday morning but received permission to skip the evening session as he headed home to Augusta, Ga., and waited to see if a deal was ironed out.
In fact, Parker has been all business.
“He has done a great job focusing on football since he came back to school,” Swinney said on the team’s practice report.
The Rockies are making quite a habit out of drafting quarterbacks from the collegiate ranks. Smooth-fielding first baseman Todd Helton and outfielder Seth Smith also were former QBs taken by the Rockies, Helton playing at Tennessee and Smith serving as a backup to Eli Manning at Ole Miss.
Even more, the club selected North Carolina State quarterback Russell Wilson in the fourth round in June.
Wilson threw 31 touchdown passes last year to lead an N.C. State offense that averaged 30 points a game.
He’s in a competition with Mike Glennon for the starting job, though, after skipping spring drills to focus on baseball.
Wilson spent the summer hitting leadoff and playing second base with Colorado’s minor-league affiliate, the Class A Tri-City Dust Devils.
The Rockies used a pick in the supplemental round to take the 18-year-old Tago. This was a compensation selection the team received for losing All-Star righty Jason Marquis to the Washington Nationals in the offseason.
Tago was 10-3 with a 2.77 ERA for Dana Hills High School in Dana Point, Calif., last year.
AP sports writers Pete Iacobelli in Clemson, S.C., and Aaron Beard in Raleigh, N.C., contributed to this story.