New to AE - currently attempting to rig a character. Plugins not allowed. How can I switch the hands of my character? The forearm is parented to the first hand already. I'm unable to parent the forearm to the new hand of my character.

Plugins like DUIK do this automatically for you, but as you say you can’t install plugins, so you’re stuck doing a manual inverse-kinematic.

As I understand it, you have keyframes on the hands and have parented a forearm to the wrong hand.  What I think you’ll need to do, is unparent the forearm.  Then parent it to the new hand.  And you’ll probably have to move the arm, its anchor and rotate it.  There might be expressions pointing to the rotation options.  This should not be a problem if you have built the rig yourself, but if you’ve been given someone else’s rig, then you’ll need to go hunting through the transform properties for each arm to alter the expressions.