It's not a paratrooper cap, it's a garrison cap with an Airborne sewn on the side. The Army had garrison caps until 2001 when they stole the black berets from the Rangers.

Well it's not Politically correct but , we used to call it a C&$t Cap! Hated the stupid thing. It is like the beret, no bill while standing in formation to keep the sun out of your eyes. You young kids don't remember Payday Friday standing out in dress "class A's" uniform for inspection!

And on a second note, people can't wear it properly any way. They don't Slant to one side of the other, PARALLEL with the ground (deck for other folks). Same with the patrol cap now days! Not a baseball cap, doesn't go down to your ears were the top is rounded by the top of the head. If it goes that low it is the WRONG size. Doesn't sit on the top of the head like a Kippah. 2 fingers off the brim of the nose and parallel to the ground!

I like the patch and have a couple of the caps that were given to me by my dads friends. That said I didn't like the garrison cap when I was a boy scout and liked it even less as a Soldier. It is in my opinion (historic value aside) even more worthless than a beret as a practical piece of head gear.
Keep the beret, so many generations of Soldiers have worn it as a symbol of distinction that it would be a shame to part with it as part of the Airborne uniform.