Как я могу заставить wget сохранять структуру подкаталога при игнорировании родительских каталогов?

Посмотрите -nH и --cut-dirs опция в wget. Из страницы справочника:

   --cut-dirs=number
       Ignore number directory components.  This is useful for getting a fine-grained control over the directory where recursive retrieval will be saved.

       Take, for example, the directory at ftp://ftp.xemacs.org/pub/xemacs/.  If you retrieve it with -r, it will be saved locally under ftp.xemacs.org/pub/xemacs/.  While the -nH option can
       remove the ftp.xemacs.org/ part, you are still stuck with pub/xemacs.  This is where --cut-dirs comes in handy; it makes Wget not "see" number remote directory components.  Here are
       several examples of how --cut-dirs option works.
           No options        -> ftp.xemacs.org/pub/xemacs/
           -nH               -> pub/xemacs/
           -nH --cut-dirs=1  -> xemacs/
           -nH --cut-dirs=2  -> .

           --cut-dirs=1      -> ftp.xemacs.org/xemacs/
           ...

   If you just want to get rid of the directory structure, this option is similar to a combination of -nd and -P.  However, unlike -nd, --cut-dirs does not lose with subdirectories---for
   instance, with -nH --cut-dirs=1, a beta/ subdirectory will be placed to xemacs/beta, as one would expect.