Rybarikova, seeded sixth, posted the only service break of the third set to take a 6-3, 4-6, 6-4 win over Parmentier. Rybarikova broke Parmentier's serve six times in 14 games in the match.
Seventh-seeded Anna-Lena Groenefeld also advanced to Friday's quarterfinals, taking out Tereza Hladikova 6-3, 6-2 in 52 minutes. She lost just seven total points in nine service games.
Yaroslava Shvedova ousted ninth-seeded Shahar Peer with a 6-2, 6-1 rout and earned a quarterfinal slot against Yvonne Meusburger, who topped Tatjana Malek 6-2, 2-6, 6-4.
Three of the tournament's top seeds, including No. 1-seeded Alize Cornet, are scheduled to play Thursday in the conclusion of the second round.
The Gastein Ladies tournament concludes Sunday with the champion taking home $37,000.