Here is the picture of timeline of Perseverance (rover) landing:
How do they make sure parachute won't fall on the rover?
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When the parachute is separated from the rover, the horizontal movement of the chute is stopped immediately. But the rover keeps moving in the same direction and with the same speed as before.
The chute has a large surface and a small weight, the rover a large weight and a small surface. So the marsian atmosphere stops the movement of the separated chute but not the rover.
The chute may fall over the rover only if there is a strong wind with just the right speed and direction to blow the chute over the rover. But such a combination has an extremly low probability.