Parakeets fighting or playing?

I don't have an answer for you without seeing the actual scene.

Yes, certain birds might be territorial. Sometimes they can simply get mad at each other. Parakeets can fight for real and I have seen it. What you described isn't clear enough for me to be sure if it was a real fight. Sometimes they do play around and it might look like fighting.

Take a video and post it will be very helpful.