The A's made the moves before opening a three-game series Friday night against the New York Yankees.
Callaspo is batting .218 with three home runs and 20 RBIs. Parrino began the spring with the A's before Texas claimed him off waivers March 3.
Parrino batted .189 in 13 games with Triple-A Round Rock when the A's claimed him off waivers April 21 and optioned him to Sacramento. He's batting .344 with three home runs and 24 RBIs in 44 games with Sacramento.
The A's also announced they have agreed to terms with 18 draft picks.