When editing parts, the Move Up and Move Down options are only available if more than one part is in the same part container. If only one part is available in a part container, then the options will be hidden.. To resolve, perform the following: 
  1. Edit the desired page In NetCommunity
  2. Locate a part container that already has at least one part present
  3. Click the cog icon on the part that is already present then click "Insert Part"
  4. Locate the desired part in the parts list, click the part then click "Select"
  5. Click the cog icon again and note that Move Up or Move Down will now be available