Parmentier had the two-time Grand Slam champion from Germany on the ropes with a break and a 4-1 lead in the third set.
Parmentier, the world number 62 from France, had the two-time Grand Slam champion from Germany on the ropes with a break and a 4-1 lead in the third set.
But Kerber fought back with a break in the seventh game and another in a marathon 11th game that gave her a 6-5 lead and a chance to serve for the match.
"I think I was starting to believe in myself again and trying to take this challenge and trying to play point by point," Kerber said of her mindset after going down a break in the deciding set.
"I was not thinking about what has happened before, just going for it and trying to move in good, bring a lot of balls back, and going for it when I have the chance."
Kerber, who rose to number one in the world with her US Open triumph last year, is set to regain the summit no matter how far she goes this week thanks to the injury withdrawal of Serena Williams.
She needed two hours and 33 minutes in the baking sun on Stadium Court to get past Parmentier, who until this year had never won a match at Indian Wells in five prior appearances.
Kerber sealed it on her third match point -- but not before Parmentier had one last chance to break in the final game.
Kerber next faces Russian Elena Vesnina, a 6-4, 1-6, 6-4 winner over Hungarian Timea Babos.
Fourth-seeded Romanian Simona Halep, yet to regain peak form in the wake of a nagging knee injury, was dispatched by in-form Kristina Mladenovic 6-3, 6-3.
France's Mladenovic, who has already lifted one trophy and reached another WTA final in 2017, had no trouble notching another top-five win.
"I perfectly played the game plan I had," Mladenovic said. "Solid at times, make her play balls and making her take some risk. Sometimes when I had the shorter (ball) stepping in, playing my heavy shots with the forehand. I think that worked pretty well today."
Halep, the 2015 Indian Wells champion, wasn't too discouraged, saying the knee trouble that forced her to pull out of St. Petersburg before her quarter-final in February had since limited her training.
"After the first set my legs were gone," Halep said. "My tennis legs were not here yet."
Seven-time Grand Slam champion Venus Williams booked a fourth-round berth with a 6-4, 6-2 victory over Lucie Safarova.
Williams, seeded 12th in her second tournament since a runner-up finish to her sister Serena in the Australian Open in January, had saved three match points en route to a second-round victory over Jelena Jankovic, but needed no heroics against the 40th-ranked Czech.
"It's never easy to play a lefty because you feel the ball coming from all different directions. You feel unbalanced, but my focus was so laser that it felt good to compete in these conditions," said Williams who belted 15 winners with just nine unforced errors, breaking Safarova four times."
The American next faces China's Peng Shuai, who ousted sixth-seeded Agnieszka Radwanska of Poland 6-4, 6-4.
American Madison Keys, playing her first event of the year after surgery on her left wrist, eased past Japan's Naomi Osaka 6-1, 6-4.