Parker was placed on the 10-day disabled list Sunday. Left-hander Steven Okert was recalled from Triple-A Sacramento in a corresponding move.
Parker's grab robbed DJ LeMahieu of an extra-base hit with two outs in the fourth inning.
After the collision, Parker sprinted toward the infield, stopped and dropped to his knees on the outfield grass. He grimaced, was examined by trainers and was replaced by Aaron Hill.
"You know you are all in when you make a play like that," Giants manager Bruce Bochy said. "You're focused. I had no idea he was going to catch that ball. To see him stay locked in was a hats-off moment to him and a huge play, it kept us in the game."
"I had no idea that ball was even catchable," said Giants pitcher Matt Moore, according to the San Francisco Chronicle. "It was smoked off the bat, and to see him just stay locked in on that, hats off to him.
"I'm terribly sorry he's going to deal with that. Hopefully it's a speedy recovery for him. It sucks when something like that happens."
Parker, 28, has appeared in nine games for the Giants this season and is batting .143 with two RBIs.
The Associated Press contributed to this report.