Member Walkalong over at THR came up with the solution to this puzzle. I'll share it here for others who might get caught by it in the future.
If a thread is locked, the attachments will not have the check boxes used to delete them. The thread has to be reopened for the check boxes to reappear. In forums other than the Gun Show forums, you will have to PM any mod to assist with deleting attachments from a locked thread.