In addition, the Reds agreed Friday to a minor league deal with catcher Miguel Olivo.
The 30-year-old Parra had spent all five of his big league seasons with the Milwaukee Brewers. After missing 2011 with back and elbow injuries, he went 3-10 with a 5.06 ERA last year in 16 starts and 26 relief appearances.
He could earn an additional $400,000 in performance bonuses based on games pitched in relief: $50,000 each for 45 and 50, and $100,000 apiece for 55, 60 and 65.
To clear a roster spot, Cincinnati designated right-hander Todd Redmond for assignment.
Olivo, 34, hit .222 with 12 RBIs in 315 at-bats last year for Seattle, his second season with the Mariners.