Most of the arguments we use to prove airplanes exist can be used to prove that dragons exist as well. We sometimes see white lines in the sky and we say they are evidence of jet airplanes. But saying they are the evidence of dragons is just as valid. There are people who say they have flown on an airplane, and use it as a proof that airplanes exist. But they could just as easily say it for dragons. And history tells us that before people claimed to have flown on a dragon just as often as people say today they have been on an airplane.
In reality, what we usually mean when we say airplane is so called jet airplane, and they can be disproven with some basic physics. Jet airplanes are supposed to work by having water (or some other liquid) as a fuel and engines forcing that water to go out, so that that water accelerates and, by the Newton's third law, makes the airplane accelerate also. But remember the Torricelli's law? Most of the people have learned it school, they just have never really thought about it. If they have, they would realize that it makes the airplanes impossible.
One of the wellknown formulations of the Torricelli's law is that, when a liquid goes through a small hole (an outlet), its speed is determined by the formula:
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v=sqrt(2*g*h)
But there is a pretty obvious implication here. That is:
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a=0
The Newton's second law tells us:
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F=m*a
Therefore:
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F=m*a=m*0=0
So, by the Newton's third law:
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F1=F2
0=F2
F2=0=0
So, the force acting on an airplane itself is zero, so by the Newton's first law:
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F=0

V
a=0
So, how can jet airplanes work in reality if they don't even work on paper? You may give me some counterexample to the Torricelli's law. But do the counterexamples matter? They don't. The Torricelli's law is derived from the Bernoulli's equation, and it's derived right from the Newton's three axioms.
Also, the burden of proof is definitely on you. You can't prove for anything that doesn't exist that it doesn't exist, but, in general, if something exists, you are able to prove it. And Occam's razor always favors more an explanation that involves someone lying or hallucinating than an explanation that involves something as complicated and as crazy sounding as airplanes.
And you might ask me what if I am wrong. So what if I am wrong? At least I am thinking about whether airplanes exist, and other people aren't thinking about that at all, they just accept what most people believe as fact. And you are way more likely to be wrong if you aren't thinking than if you are thinking.
(This is my parody of the Internet conspiracy theorists!)