I find it totally non-intuitive that a parachute would jump back a little when deployed. I am not talking about the effect as seen by other watchers. Is it possible that some air is compressed under the chute such that it causes that small bounce back?
If you're asking what I think you're asking, then your answer lies within the falling of the person. When someone is falling through the air, the wind is rushing up relative to the person because the person is moving down relative to the air. In other words, what causes the "blowback" is the drag force suddenly increasing many times over. Drag is the force something (or someone) experiences when trying to move in a fluid. It can be said to be a complicated type of friction.
In this case, a parachute works by drastically increasing the amount of air a person has to push out of the way to move in it. That increase is due to a few things but mostly it's the increase in cross-sectional area and changing of the drag coefficient of our object (in this case our skydiver and parachute).
If you're asking about myth number 8 in your link, I believe the effect your link describes is the relative "going up" of someone who has deployed a chute as seen by someone who has not yet deployed their chute. This is merely due to the sudden change of velocity of the person who deployed the chute.
Think of it this way; the two people skydiving are two drivers accelerating side by side towards something. One of them brakes, hard. The other driver who is still going at the original high speed sees the car next to him "bounce back" but to someone standing still looking at both of them, the car just slowed down, it didn't go backwards. Similarly, the parachute deployer only slows down, he/she doesn't go up. It only looks that way to the other person falling down.
Edit: To answer your comment's follow-up, no. This is because of Newton's laws. In much the same way that you can't pull yourself up by your own bootstraps, you can't fall fast enough or deploy a parachute immediately enough for the air to do that (assuming the air is static). The compression of air is due to gravity and the going up that you ask about fights gravity. There is no way to obtain a bigger force going up from a force going down, no matter how compressed the air. It's like if you had a weight on top of a spring and tried to make it bounce higher than its release point, it's not going to happen. It may happen in real life but if it did, then it's because of air currents.