I haven't taken paralysing palm yet but in the description it says you "must be completely unarmed" to use it. Does that mean you can't even have an unarmed weapon equipped such as spiked knuckles or a ballistic fist equipped to use it? What about armor? Just wanted to know before I took it.

No weapons. Just your fists.

Corrected: http://fallout.wikia.com/wiki/Paralyzing_Palm: It turns out that it does work with weapons.

I do wonder if it works with upper cut

Wrong, it works with unarmed weapons as well.

100% certain that I scored a Paralyzing Palm while wearing a Ballistic Fist.

Whether that is a bug or intended....dunno. But you can definately use the perk while wearing Unarmed category weapons.

Paralyzing Palm works with unarmed weapons but the palm strike animation only occurs if you are not using any weapon.

Works with unarmed weapons. My favorite perk in the game.