Your statement, "If there's no air moving over and under the wings, no lift can be generated" is of course true, but the treadmill would not cause that condition to exist. On the ground and in level or climbing flight (and to a certain degree descending flight under power), airplanes move forward due to air being forced rearward. Once they reach a certain speed, the wing starts to develop lift because of the difference in airflow speeds above and below the wing. Air travels faster over the top of the wing, developing an area of lower pressure than below the wing... this is "lift." Groundspeed (and airspeed to varying degrees) is determined by the air being forced backwards, and increased as the airplane overcomes inertia and gains momentum. As the speed increased and lift develops, the airplane gets lighter so there's less drag at the wheels/ground, speed continues to increase, and the airplane lifts off once it generates more lift than it weighs. Again, there's more to it than that, but that's a pretty good basic way to look at it.
Back to the treadmill and your human-w/-a-parachute scenario. If the human was walking or running on the treadmill and was not going forward in real space, then of course his parachute would not inflate; it would be stationary. BUT... if the person on the treadmill had on roller skates or was on a skateboard and had a big ol' powerful fan strapped on in such a way that airflow was unimpeded (and the treadmill was very very long, like one of those moving walkways in an airport), they'd develop forward motion no matter what. If the treadmill was going so fast that the friction in the wheel bearings of the skateboard heated up and seized, preventing the wheels from turning faster than the treadmill, then you'd be correct. As long as the skateboard's wheels spin freely, the treadmill is immaterial... our hero will move forward just as he would on non-moving surfaces due to the air being forced behind him.
Stand outside in a strong wind and face the wind. You'll find yourself leaning forward, like you would if you were standing on a very steep hill, even though you're on level ground. Turn so the wind is at your back, but stand in the same place. Now you have to lean backwards so you don't get blown forward or fall over. The ground hasn't changed, yet your "attitude" is different. The wind doesn't care about the ground (again, vast oversimplification... ground effect, airflow over obstacles, blah blah blah.. but for our purposes, we can ignore that stuff for now). Nor will the airflow over the airplane be influenced by the treadmill. The ONLY way for a treadmill to have any influence on an airplane's speed is if the airplane's WHEELS are prevented from spinning completely freely.
Let me try one more.. even simpler. Stand outside on the road (watch out for cars) on a very windy day. You won't go anywhere. Now, stand on a skateboard in the same place. The wind will blow you down the road. The speed of the road surface hasn't changed; you've just lowered the friction component between you and the road (your feet on the road vs. the skateboard's wheels on the road) so you move down the road under the force of the wind.
You are right about the airflow over the wings being necessary to produce lift. It doesn't matter how that airflow is produced. If the airplane was completely stationary but a fan was generating airflow over the wings at a high enough velocity, the airplane would still lift off. There's a REASON we "tie down" airplanes when parked outside hangars; airplanes can be and ARE knocked over and even spontaneously relocated by very strong winds.
In real world physics, there's probably a limit as to just how fast the wheels on a given plane COULD turn before creating enough heat to burn out the bearings and seize up. If you could somehow create an airplane-sized treadmill capable of whatever that speed is (so already we're leaving the real world again) and you could cause that condition before the airplane took off, then I guess you could win the argument. It would have to happen very quickly, because no matter how fast the treadmill was moving, the airplane WOULD start to move forward the instant the prop starting pushing air rearwards.
One more point - airspeed and groundspeed are two very different things. Airplanes can hover over a point on the ground. W/ full flaps down, my plane can fly as slow as 40mph. If I fly directly into a 40mph headwind, my net forward progress is 0mph, yet there's enough airflow caused by my prop and the headwind to generate enough lift to hold altitude.
Somebody nudge @
Ultramaroon ... I think he nodded off after the first sentence...
Barry