The moves came after Tuesday night's 5-2 loss to the Washington Nationals.
Clement batted .189 with a homer and seven RBIs in 45 games for the Pirates. He started 30 games at first base but was in a 1-for-17 slump.
Taschner was 1-0 with a 6.05 ERA, eight walks and 22 hits allowed in 19 1/3 innings. He was reinstated from the 15-day disabled list Wednesday after missing time with a strained left hamstring.
Pirates manager John Russell said the team will announce more moves Wednesday to fill the roster spots.