Ortiz went 4 for 5 with two homers and four RBIs, but it wasn't enough.
Ortiz hit solo homers in the first and fourth and is tied with Jim Rice on the Red Sox list with 382. It was the 42nd multihomer game for Big Papi, who began his career with the Twins.
Ortiz added an RBI single in the sixth and another one in the seventh that made it 6-5.
Boston scored three times in the seventh and tied the score at 6 after reliever Matt Guerrier walked pinch-hitter Jonny Gomes with the bases loaded.
Leading 6-3, Minnesota left the bases loaded in the sixth. It also stranded runners on second and third in the third and fifth.
Red Sox starter Jake Peavy allowed six runs on nine hits in 4 1-3 innings and failed to complete at least five innings for the first time in 18 starts with Boston.
The Minnesota Wild goal horn blared every time the Twins scored a run as the Wild played the Chicago Blackhawks in the NHL playoffs a few miles down the road in St. Paul.
As soon as Parmelee touched home plate, the horn blared even louder as Parmelee's teammates mobbed him all the way to the dugout.
A victory would have given the Red Sox their first three-game winning streak of the season.
Boston manager John Farrell originally listed Peavy as Wednesday's starter, but said he intended to start Peavy on Tuesday all along and ''screwed up'' when announcing the probables for this series.
NOTES: Twins GM Rob Antony said outfielder Josh Willingham, out since April 7 with a broken wrist, took early batting practice and could begin a minor league rehab assignment this weekend. ... Antony also said an MRI showed a re-aggravation of the wrist injury sustained by top prospect Byron Buxton late in spring training. Buxton will work out in Fort Myers, Florida, until the wrist feels better. ... Peavy has allowed a home run in nine straight regular-season starts. ... Farrell said before the game that Twins manager Ron Gardenhire and Indians manager Terry Francona will be his coaches for the All-Star game at Target Field in July. ... Felix Doubront (1-3, 5.09 ERA) originally was scheduled to start Tuesday for Boston, but was given an extra day of rest and will start Wednesday against Kevin Correia (1-4, 6.34).