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.
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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)
