The injury means 36-year-old Jose Valverde is likely to take over as closer for now in an already thin bullpen. Valverde, a three-time All-Star, has struggled with control in recent years.
Parnell gave up Denard Span's tying double in the ninth inning during Monday's opening 9-7 loss to Washington. It was the right-hander's first save opportunity since a neck injury that required surgery and sidelined him for the final two months last season.
He complained of tightness in his right elbow after Monday's game. The Mets said the 29-year-old 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.
New York plans to bring up Kyle Farnsworth to take Parnell's roster spot before Wednesday's game against the Nationals. Infielder Wilmer Flores also is likely to be recalled from the minors to replace second baseman Daniel Murphy, who would go on paternity leave.
Parnell was 5-5 with a 2.16 ERA and 22 saves in 26 chances last year. He averaged 92.1 mph for 20 two-seam fastballs during Monday's outing, according to fangraphs.com, down from 94.9 mph last year and 96 mph in 2012.
New York said he received a plasma injection and will rest about two weeks. Parnell then will start a throwing program, and a decision on surgery will be made after that.
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.
Valverde has 286 saves in an 11-year big league career with Arizona, Houston and Detroit. He was 0-1 with a 5.59 ERA for Detroit last year, converting nine saves in 12 chances before he was cut from the big league roster in mid-June 21. He then had seven saves with a 4.09 ERA in 11 games at Toledo and was released Aug. 7.
He got four straight outs Monday in his Mets debut, striking out Ryan Zimmerman, Jayson Werth and Bryce Harper.