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.
"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.
"It was a nice little ballgame again," Twins manager Ron Gardenhire said. "[Pitching coach Rick Anderson] and I were talking, and pitching wasn't as sharp as yesterday. There were a few guys that were sharp, but we were up in the zone quite a bit. So that was just one of those days. It was our first day at home, so our guys might've been overthrowing a bit."
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: Right-hander Ricky Nolasco is set to make his Twins debut against the Rays in Port Charlotte on Sunday at 12:05 p.m. CT. Nolasco joined the club this offseason on a four-year, $49 million deal. Right-hander Alex Cobb is set to start for the Rays in the first meeting between the two teams this spring at Charlotte Sports Park.
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.