You'd always say "my father is a pilot". It's unnatural to say "He flies airplanes" as saying he is a pilot already implies he flies a plane. If your father flew something else then you'd specify what he flew. For example if it were a helicopter, you'd say "My father's a helicopter pilot".

What fun, anyway...flying planes!!!

He flies airplanes (simple present). That's the tense you would use for actions that happen frequently. For example, "I teach English" or "I drive to work everyday" or even "I see my grandparents once a year" (How frequent the action is, is not important). "He is flying airplanes" is incorrect because it's in present progressive which is used for actions that are happening at the moment, such as, "I'm typing right now". So, even if you wanted to mean your father is flying an airplane at the moment, you wouldn't say it in plural, because that would imply that he is flying more than one plane at a time.

He flies airplanes is correct as it means he flies airplanes regularly. He is flying would mean he is flying right now and it would be He is flying AN airplane as he can't pilot more than one airplane at the same time. Present simple tense versus present continuous tense

My father is a pilot.