The Mets said Parnell received a platelet-rich plasma injection in his elbow and will rest for approximately two weeks before undergoing further evaluation, when the determination will be made whether he will have surgery.
The injury means 36-year-old Jose Valverde is likely to take over as closer in an already thin bullpen.
Parnell complained of tightness in his right forearm after Monday's season-opening loss to the Washington Nationals. The Mets said he was examined Tuesday at the Hospital for Special Surgery, had an MRI and was diagnosed with a partial tear of the right medial collateral ligament.
He blew a save opportunity in Monday's 10-inning loss, squandering a one-run lead in the ninth inning. It was his first save opportunity since a neck injury that required surgery and sidelined him for the final two months last season.
The Mets will promote veteran reliever Kyle Farnsworth from the minors to replace Parnell, who will be placed on the disabled list.
Parnell's velocity sagged throughout spring training as he attempted to bounce back from a herniated disk. The 29-year-old had a 2.16 ERA and 22 saves last season in his first full-time stint as New York's closer.
Valverde logged 1 1/3 perfect relief innings in a setup role Monday, striking out three. The three-time All-Star had nine saves last year with the Detroit Tigers and has 286 career saves over 12 seasons.
The Mets already are without sidelined ace Matt Harvey, who is likely to miss the entire season following elbow surgery Oct. 22 on a partial tear of his right ulnar collateral ligament.
Several pitchers have torn elbow ligaments this year, including Atlanta's Kris Medlen and Brandon Beachy, Arizona's Patrick Corbin, Detroit's Bruce Rondon and Oakland's Jarrod Parker. Braves pitcher Cory Gearrin is being examined for a ligament injury.
ESPNNewYork.com's Adam Rubin and The Associated Press contributed to this report.