So after revisiting this, I tried l0b0's method. It comes back with:
tar: Options `-Aru' are incompatible with `-f -'
So I looked into another way and came up with this script:
#!/bin/bash
find /volume1/Media/TV\ Series/ -iname "*.sub" -exec tar rvfP /volume1/Backups/TestBackup.tar {} \;
find /volume1/Media/TV\ Series/ -iname "*.srt" -exec tar rvfP /volume1/Backups/TestBackup.tar {} \;
find /volume1/Media/TV\ Series/ -iname "*.nfo" -exec tar rvfP /volume1/Backups/TestBackup.tar {} \;
find /volume1/Media/TV\ Series/ -iname "*.jpg" -exec tar rvfP /volume1/Backups/TestBackup.tar {} \;
find /volume1/Media/TV\ Series/ -iname "*.tbn" -exec tar rvfP /volume1/Backups/TestBackup.tar {} \;
gzip /volume1/Backups/TestBackup.tar
It takes a long time, but I let it run in the middle of the night when nothing else is going on, and it works. I only need to run this backup very infrequently anyway.
If you were so inclined, you can also add an elapsed timer:
#!/bin/bash
date1=$(date +"%s")
# Script stuff here. Uncommented...
date2=$(date +"%s")
diff=$(($date2-$date1))
echo "$(($diff / 60)) minutes and $(($diff % 60)) seconds elapsed."