The injury means 36-year-old Jose Valverde is likely to take over as closer 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.
Texas pitching coach Mike Maddux said the right-hander looked good during what amounted to four innings of simulated work at the Rangers' home ballpark Tuesday.
Maddux says the next step is to see how Darvish recovers. He missed the opening-day start after developing a stiff neck about two weeks ago in spring training. Rangers General Manager Jon Daniels said Monday that Darvish could return without making a rehabilitation start. He's eligible to come off the disabled list Saturday.
The Chicago Cubs agreed to a minor league contract with former 16-game winner Joel Pineiro.
Pineiro won 104 games from 2000 to 2011 with Seattle, St. Louis and the Angels.
He won a career-high 16 games for Seattle in 2003. Pineiro has not pitched in the majors since 2011 with the Angels.
ed in the majors since 2011 with the Angels.