Up next for 32-year-old Parmentier is Romania's Irina-Camelia Begu, who battled past Donna Vekic 3-6 7-6(1) 6-1.
After being broken in the first game of the opening set, Wozniacki took a 4-1 lead before her opponent drew level. The Dane then switched gears to claim the opening set.
Seeking her maiden victory over Wozniacki, Parmentier raced ahead 3-0 in the second set. Wozniacki took a medical timeout to deal with her injury but struggled thereafter and called it quits after Parmentier won the second set.
Earlier, twice grand slam champion Svetlana Kuznetsova made 21 unforced errors in her 6-1 1-6 7-5 defeat by Slovenia's Polona Hercog.
Hercog, 27, will take on Maria Sakkari of Greece who beat Dutch qualifier Arantxa Rus 6-3 7-6(6).
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Pritha Sarkar)
