Parmentier fought off seven break points before beating Tathiana Garbin of Italy 6-2, 7-6 (3), while Benesova rallied to beat Kirsten Flipkens of Belgium 6-7 (5), 7-5, 6-4.
Benesova converted seven of 18 break points, while Flipkens had seven aces.
In the second round, Benesova will play Galina Voskoboeva of Russia, who rallied to beat Sanda Mamic of Croatia 4-6, 6-4, 6-0.
Austria's Melanie Klaffner, ranked 320th, upset Lucie Safarova of the Czech Republic 6-3, 6-4.
The 18-year-old Klaffner faced only two break points and held serve throughout, while Safarova double-faulted nine times.
Klaffner will play either top-seeded Agnes Szavay or Viktoriya Kutuzova in the next round.
Maret Ani of Estonia also advanced, beating ninth-seeded Klara Zakopalova of the Czech Republic 7-5, 4-6, 6-3.
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