Well, first of all you want to have your joint hierarchy set up properly before skinning. Makes your workflow much cleaner that way. Whenever you use "move skin joint tool" you do not need to unparent the joint you want to move, you just simply move it anywhere you need to.

Also, the joint you want to make as a parent, is that skinned to the mesh as well? Lastly, all you have to do is click the joint you want to be the child, SHIFT click the joint you want to be the parent and click "P", tell me if the issue remains that way, it shouldn't though!

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