Bammer will face Julia Goerges, who outlasted Maria Elena Camerin of Italy 5-7, 7-5, 6-1.
The 267th-ranked Evtimova reached the second round of a WTA Tour event for the first time in her career and will play top-seeded Maria Sharapova.
Bammer led 4-0 in the final set but faltered as Parmentier won five straight games to get an opportunity to serve out the match. But a double-fault from the Frenchwoman helped Bammer even it 5-5.
Barrois was scheduled to play fourth-seeded Peng Shuai, but Peng experienced dizziness and withdrew.