Can I build a remote controlled airplane with parts from a remote controlled car?

yes, if you get it right, however the motor may be too heavy for the plane
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With some of them. You'll need shock-absorbing landing gear (piano-wire legs will do if it's a small plane) and some mechanical gear to turn the steering into something that will move the rudder. You'll also need another channel for the elevator. You can't control a plane without at least rudder and elevator controls. Ailerons would give you the ability to do some other maneuvers. And a car motor might not spin fast enough to give you enough speed to get the lift you need, you might have to use a gas engine.

You'd probably spend about as much as if you just bought an RC plane kit, or a plane, an RC setup and an engine. (We used to start with balsa and paper, and build our own planes and our own RC gear. About all we bought were electronic components, a couple of servos [although you didn't even need that if you used a couple of motors and a "galloping ghost" setup for rudder and elevator] and a gas engine to turn the prop [which we carved out of balsa].)
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Large scale - maybe.

The servos and motors in remote cars are too heavy for most model aircraft. I know; I tried to cheap out building one once. It crashed - badly.

Learned a wealth of information about CG and weight limits that day.
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First off, you could not legally fly your creation because model airplane and surface vehicle radios are on different frequencies, also called "channels". That is so you could not endanger someones life by another surface vehicle on the same channel turning on while you are flying. It would cause you to lose control and crash, perhaps doing bodily harm.
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Maybe. The only thing is you need something to make the wing and propeller out of and the motor needs to be strong enough to pull the airplane into the air.
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