I didn't know there was more evidence needed since I am 100% sure it's Rod Allen and the comparison photo seemed spot on. When I first saw the pic in the Vault images, I said to myself "I see ya, Rod Allen." (Anyone who watches Rod as a Tigers analyst will be familiar with that reference.)
Anyway, here is a screenshot of the infielders and outfielders from a White Sox 40-man roster from the spring of 1981. It was published in the Sporting News in the March 28, 1981 edition on page 21. Rod is listed, but Lorenzo Gray is not (although Lorenzo could have been a non-roster invitee).
Also the link to an archived UPI story references Rod Allen in camp for the Sox as spring training started in 1981.
The Chicago White Sox open their exhibition season today... - UPI Archives