The 13th-ranked Mladenovic constantly pressured Flipkens' serve, taking five of 11 break-point chances and beating the Belgian for the second time in two career meetings.
Parmentier, ranked 91st, dropped her serve six times in a one-sided contest.
Mertens will face Mladenovic, and Parmentier follows on court against the 70th-ranked Flipkens in Sunday's reverse singles. The French pair will team up in doubles against Belgian duo Ysaline Bonaventure and Alison Van Uytvanck.
France is without Alize Cornet, who has not been selected after she missed three out-of-competition doping tests in the past year. She is under investigation by the International Tennis Federation. France is also missing seventh-ranked Caroline Garcia, who is still declining to play for France as she gives priority to her singles career.
France and Belgium are meeting for a fifth time, with France leading 3-1 overall and Belgium winning their last meeting in 2000. The winner faces the United States or the Netherlands in the semifinals in April.
France, which lost 4-1 to Switzerland in last year's quarterfinals, won the Fed Cup in 1997 and 2003. France has lost three times in the final since then, most recently in 2016 when Garcia was playing.