You don't have an odd problem, you have an odd workflow. Try making the layers 3d, adjusting the anchor points and animating the rotation. There are expressions that can be used to make this type of animation easier by setting up inverse kinematics for the characters limbs. Without seeing at least a screenshot of your comp I can't give much more advice. There are just too many ways to do what you're trying to do.
In case you're interested here is a link to Dan Ebberts Inverse Kinematics solution.
Is there a way to remove certain aspects of a parent's animation from a child layer?
You cannot. Somehow you you have your logic backwards. I agree with Rick on that one. As a matter of fact expressions would solve all your issues easily. Still, there's one way to steal your cake and still eat it: Rather than trying to use a layer's native transform properties, use the distortion effect of same name (or other suitable ones for that matter). Takes care of the whole problem in one bold stroke...
Thanks for your answers here. The problem is, my workflow is constrained by our game development cycle where I work. We're using code to export my AE files, and Psd layers, to animate characters in game. The issue is, we can't export any complex things like effects or bones, and probably not 3D layers. I have to stick to basic transforms right now for our code to work properly, otherwise I'd definitely try these suggestions.
I'll check out the Inverse Kinematics link, that's awesome. I only know After Effects for character animation right now, so anything like that helps me a great deal. Thanks!