Parmelee bashed a three-run homer to trigger a 10-run fifth inning, during which the Twins sent 14 men to the plate, banging out eight hits while collecting three walks.
Escobar -- a former White Sox player making his Twins debut -- delivered a game-tying, two-run single during a seven-run second inning that erased an early 3-0 Chicago lead.
Doumit, meanwhile, socked a solo homer in the seventh as Minnesota pounded out 17 hits en route to evening their three-game series with the American League Central-leading White Sox.
Scott Diamond (11-6) earned his first win in nearly a month, surrendering four runs on seven hits over five innings.
Chisox starter Jose Quintana (5-4) was rocked for seven runs on five hits in just 1 1/3 frames. Chicago reliever Phil Humber, meanwhile, was decimated for eight runs in just one-third of an inning.
Dewayne Wise drove in a pair of runs for the White Sox, who set a club record for most doubles in a game (10) in the loss.
Dennis Rodman pledges to end trips to North Korea
Jessica Simpson shares three-way kiss with friends in photo