The Rangers picked him up on March 3 but then designated him for assignment on Friday after he batted .189/.271/.245 with no homers and five RBIs for Triple-A Round Rock. Parrino spent the 2013 season in the A's organization and played 14 games with Oakland.
"We do seem to maybe target the same type of players," Melvin said. "I'm not part of the front-office dynamic as far as claiming players and so forth. But a guy that we've had, we did have a 40-man roster spot open and a guy that we know a lot of about, so it made some sense to claim him."
The move is reminiscent of the situation with infielder Adam Rosales last year, who shuttled back and forth between the two AL West foes on the waiver wire.
Alex Espinoza is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.