If you use your own account to access your own files, you will always have full-access permission. In your case, you shared a folder with a sub-user, and you shared it with Add/Edit permission. You need to share it with full access permission for the sub-user to delete files in the shared folder. You can edit the share and change the permission level. Please look at the screenshot below: (Click Manage Shares first, then click the share name)
