Petkovic slumped to a 6-4 6-2 loss in a contest in which service holds proved at a premium.
World number 16 Petkovic dropped serve in the opening game of the match and then let a 3-1 lead slip as Frenchwoman Parmentier won 11 of the next 14 games to seal victory, which she capped with an ace.
There was more success for French fans to celebrate as second seed Alize Cornet overcame Czech qualifier Lucie Hradecka 6-2 7-6 (8-6) and set up a second-round clash with Kiki Bertens following the Dutchwoman's three-set win over Anna-Lena Friedsam.
"I had a good first set; it was a bit more complicated in the second set," Cornet said.
"I got a bit tired, and Lucie started playing better, so finally at the end I missed three or four match points at 5-4, and she had set points herself, so I'm really glad I could pull it off in straight sets."
While Petkovic, Friedsam and Julia Goerges – who lost 6-3 5-7 6-2 to American fifth seed Varvara Lepchenko – all tumbled out, Sabine Lisicki and Annika Beck restored some German pride with comfortable triumphs.
Third seed Lisicki dealt a second-set bagel to Slovakian Daniela Hantuchova, winning a potentially tricky match 7-5 6-0, and last year's beaten finalist Beck was a 7-6 (8-6) 6-3 victor over Timea Bacsinszky of Switzerland.
Belgian seventh seed Kirsten Flipkens joined Petkovic in making an early exit, succumbing 7-6 (7-5) 6-1 to Johanna Larsson of Sweden to mark the third straight tournament in which she has lost her opening match after doing likewise at the China Open and the Generali Ladies Linz.
Elsewhere, there were wins for Denisa Allertova, Polona Hercog and Alison van Uytvanck.