Parmentier broke Schmiedlova five times and didn't drop her serve. She will next face Naomi Broady of Britain.
Also, seventh-seeded Timea Babos of Hungary beat lucky loser Jesika Maleckova 6-3, 7-5 to advance.
In the second round, third-seeded Jelena Ostapenko of Latvia beat Kristyna Pliskova of the Czech Republic 6-3, 6-4, and fifth-seeded Camila Giorgi of Italy came from a set down to beat qualifier Ekaterina Alexandrova of Russia 4-6, 6-2, 6-4.
Giorgi will play ninth-seeded Kirsten Flipkens of Belgium, who defeated Andreea Mitu of Romania 6-1, 6-2.
Eighth-seeded Dominika Cibulkova of Slovakia made the quarterfinals by beating Elizaveta Kulichkova of Russia 6-2, 7-5.