The White Sox will receive cash considerations in the deal. Elmore was designated for assignment by Chicago on Wednesday after he was previously claimed off waivers from the Astros on Nov. 13.
The 26-year-old Elmore has appeared in 82 career Major League games, batting .223 with 17 runs, eight doubles, two home runs and 13 RBIs. He's a .291 career hitter with 110 stolen bases over six Minor League seasons with the White Sox, Astros and D-backs.
The bulk of his big league playing time came with the Astros last season, when he hit .242 with four doubles, two homers and six RBIs.
Parrino, acquired by the A's in a 2012 trade with the Padres involving pitcher Tyson Ross, hit .118 with two doubles and one RBI in 14 games with the A's last year.