Can't find parent "With Automatic Weights" in Blender for a Unity3D .fbx import?

Question

I have no errors so I don't know what's going on. I have a fully rigged humanoid 3D model that I imported with .fbx into Blender and I'm trying to separate the arms while keeping them rigged so I can import the arms on their own back into Unity3D (with the full body still being there too separately).

I clicked on the mesh alpha_surface -> edit mode, highlighted the vertices of both arms, then P -> selection, then I get alpha_surface001.

I checked the vertex groups of the new arms, all the right bones are there, and in the armature under parent, it shows the main armature object so it all appears correct. Weight Paint mode shows both arms as fully blue, but each individual bone that I select in the vertex group is fully red.

I click on Alpha_Surface001 then shift+click on the main Armature object at the top, then do ctrl + p and no weight option comes up, only "Set Parent To". But it appears that the new Alpha_Surface001 is already a child I think so not sure what to do here unless nothing else is required.