Trevor Plouffe drew a bases-loaded walk to bring home a run before Kennys Vargas added a sacrifice fly to left to give the Twins an early 3-1 lead.
Left-hander Scott Diamond started for Minnesota, giving up one run over two innings on a solo shot by Daniel Nava in the first inning. Diamond walked Mike Carp to open the second inning, but was able to get out of the jam.
"Those first couple of swings weren't too good, so I was trying to shorten things up," said Nava. "Fortunately, he threw a good pitch -- and fortunately the ball went over the fence."
"The Nava home run, those are going to happen," said Diamond, who is competing for the fifth spot in Minnesota's rotation. "It's just upsetting it happened the first outing. But the walk is what I was most upset about. It's never what you want to do. But I was able to attack the rest of the inning and get out of it, scot- free."
The Red Sox scored again in the third inning on an RBI single by Jonathan Herrera to score Brock Holt, who led off the inning with a single off Twins right-hander Ryan Pressly.
But the Twins broke it open in the fifth inning on a three-run blast by Parmelee off left-hander Jose Mijares. Mijares, who played with the Twins from 2008-11, recorded just two outs, giving up four hits and a walk, including Parmelee's homer.
Up next: Lefty Felix Doubront will be the first member of Boston's projected rotation to get a start this spring, when he pitches Sunday's 1:05 p.m. ET home game against the Orioles. Edward Mujica, expected to be a key piece in the bullpen, will see his first action, as well. David Ortiz, Will Middlebrooks, Xander Bogaerts, Mike Napoli and Dustin Pedroia should all be in the lineup.
Rhett Bollinger is a reporter for MLB.com. Read his blog, Bollinger Beat, and follow him on Twitter @RhettBollinger. This story was not subject to the approval of Major League Baseball or its clubs.