You can either use the Vertex Weight Edit modifier or, more simply in this case, the Vertex Weight Mix modifier:
Which will let you subtract the band group from the circle group:
Ensure 'Use Modifier Stack' is checked in the particle system..
Your question sounds like you have two different armatures (which are objects), you're selecting both and then entering edit mode (which in this case will be a "double" edit mode, working simultaneusly on two different objects). But the parenting you're looking for is possible only internally to one single object.
So, select the eye armature, shift select ...
You need to parent the mesh to the armature first to setup the weights, this is done with Lock Object Modes checked.
In Object Mode, select the mesh, Shift select armature, then Ctrl + P to parent and choose Armature Deform then select any sub-option, like With Automatic Weights.
You should now be able to Ctrl + left mouse on the bones in weight paint mode ...