I have 3 separate armatures, A, B, and C. I want to parent one bone from Armature A and another bone from Armature B to one bone in Armature C. I'm trying to make one bone in Armature C offset the other two bones in Armatures A and B whenever it is moved in pose mode. Is this possible? If so, what is the process for setting this up?
Yes, from Pose Mode in the Bone Constraints tab you can select the bone you want to act as the child and add a "Child Of" constraint. Point it to the armature, then within that the bone, that you want it to copy transformations of.
You'll probably need to uncheck the X rotation transformation, because bone axes are wonky (that's a technical explanation). Maybe someone else can explain why, but I've just come to accept it as the way it is. In any case, this works.
There's a method of constraining that works especially well with bones, and avoids unwanted offsets caused by double transforms.
go to pose mode, select the child bone, add a "copy transform" constraint, set it to Parent armature, parent bone, local space both. The same for the others child bones.
Consider that you can even animate the influence of the constraints.
Another way is using drivers, which are more powerful (you can program curves of influence and many others complex behaviours) but more difficult to explain in a few words; search for info about them, even in this site.