The air does indeed accelerate together with the Earth, being pushed up by it. The effect of the parachute is generated by the air resistance or drag.
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Let's look at this from the bigger picture down to the smaller.
Gravity on the FE is supposedly the effect of the earth rushing upward at some constant speed, keeping everything pinned to the ground.
Once a person is free from the surface of the earth, through whatever means (but we'll go with an air plane here), they are free from the effects of gravity and are actually waiting for the earth to catch up to them, since there supposedly no gravity and only the effect of the earth rushing up.
Now, if a person jumps out of an air plane, they would not actually be falling but would remain stationary (at the same altitude of the plane they jumped from) waiting for the earth to catch them because, as we've seen, gravity is an effect of the earth rushing up.
Now let's look at the parachute question/answer.
First, since gravity doesn't exist off the surface of the earth, there would be no need for a parachute.
Second, even if the person who jumped from the plane started falling in a zero gravity environment, the parachute would cause a drag situation. How does the drag provided by the parachute account for the force of the earth, rushing up to and hitting a person at 21 miles per hour?
A two ton (1,814 kg) vehicle moving at 21 miles per hour can do considerable damage to the human body, what about a multi million ton earth?
The following is based upon physics within the realm of reality.
An average person weighing 80 kg, falling from 5 meters, at a velocity of 9.8 meters per second, hits a surface with an average force of 39,200 Newtons.
A two ton (1,814 kg) vehicle traveling at 9.8 meters per second, through 5 meters of distance (taken to be analogous to falling), hits a surface with an average force of 1,778,081,620 Newtons.
The Earth, weighing 5,972,000,000,000,000,000,000,000 kg traveling at 9.8 meters per second (also taken to be analogous to falling), through a distance of 5 meters, hits a surface with an average force of 2.9262799999999995e+27 Newtons.
What would actually happen to that unfortunate soul, hanging from the end of a parachute, when the Earth strikes him? Imagine all that force being applied to an area of a few square inches (the feet) and being transferred upward through the body by all the various bones from the feet leading up to the skull. Do you think the person would feel it before they died?