Are female humans allowed to fly aero planes?

Nope!  Only female bonobo chimpanzees can do that!

Yes. 

They are. 

As private pilots, commercial pilots and military pilots.

Female pilots fly all sorts of aircraft including military and airline.

 I've held a private pilot's license longer than I've had a driver's license.

 I was at the controls of my dad's plane at age 12 and was licensed at age 20. The driver's license came at 22. 

 When Dad died I inherited his well aged Cessna 180. I sold it recently due to the exorbitant amount of money it was going to cost for and engine rebuild and a wing spar corrosion issue. I have regretted that choice ever since.

Watch this all female crew fly the cargo MD-11 from Frankfurt to Mumbai.  Somewhere in the video they tell how they become pilots:

https://www.youtube.com/watch?v=bzsGG-SPkVs

Ever heard of Amelia Earhart?

The answer is yes.

Of course, even in the airforce. Some females fly large passenger and large freight planes also.