Konta struggled on Friday during her 6-2, 6-7 (5), 6-4 win over Aaksandra Sasnovich in the second round but she looked back to her best on Sunday.
Parmentier had arrived in Miami full of confidence after reaching the third round in Indian Wells where she took a set off second seed Angelique Kerber.
But after an even start to the match Konta seized the initiative in the seventh game of the match when she swatted away a backhand to take the first break.
With a first serve percentage in excess of 90 per cent, Konta gave her opponent no way back in the opening set and would step up her advantage in the second.
Immediately, Konta broke to take the first game of the second set then a superbly measured forehand into the corner gave her the double break for 3-0 and effectively sealed a surprisingly routine success.
Growing increasingly disillusioned, Parmentier's game collapsed as Konta broke again to love before duly serving out to take her place in the last 16.
She will now face the winner of the match between Lara Arruabarrena and Madison Keys.
In the men's doubles Jamie Murray and Bruno Soares are through to the quarter-finals following a 6-0, 6-3 win against Paolo Lorenzi and Joao Sousa.
Additional reporting by PA.
Read the original article on Eurosport: Konta eases into fourth round in Miami