The White Sox allowed 32 runs in a doubleheader
Using Baseball-Reference's play index, which only encompasses the last 50 years, the worst single-day runs allowed output I can find is 25 runs on August 30th of 1970 [Game 1 - Game 2]. I think I heard Bill Melton say something about the Sox record for runs allowed in a DH was 27, but I have no clue how to query Retrosheet's database to find out for sure. At any rate, this was the single worst day of White Sox pitching in my lifetime.
It was the first time in 8 years that the Sox have allowed 12 runs or more in back-to-back games. 8 years ago, they did one better, allowing 12 or more in 3 straight to the Indians [Game 1 - Game 2 - Game 3], which gives them something to shoot for, I suppose. (There's some great names in those box scores. Just looking at the losing pitchers speaks volumes; Navarro, Baldwin, Parque... Yuck!)