The short answer is—believe it or not—that an airplane is “allowed” to take off in a thunderstorm—if the pilot is bold (read: stupid) enough. Your use of the word “allowed” suggests that the airport's control tower can dictate a pilot's actions as regards weather. No. Their job is traffic control. read more
Taking off with thunderstorms in the vicinity of the airport is almost routine and is acceptable, as in the video. As I did many times with thunderstorm cells around the airport, before taking off I would sit at the end of the runway. I would then wait for the storm to blow over and pass. read more
An airplane will never ever fly through a thunderstorm on purpose. The lightning inside a thunderstorm is not the factor that makes this a general rule in aviation. Basically a thunderstorms consists of constant updrafts and downdrafts, and the borderline between these create wind shear. read more