youtube-dl --help, contains:
Video Selection: --playlist-start NUMBER Playlist video to start at (default is 1) --playlist-end NUMBER Playlist video to end at (default is last) --playlist-items ITEM_SPEC Playlist video items to download. Specify indices of the videos in the playlist separated by commas like: "--playlist-items 1,2,5,8" if you want to download videos indexed 1, 2, 5, 8 in the playlist. You can specify range: "--playlist-items 1-3,7,10-13", it will download the videos at index 1, 2, 3, 7, 10, 11, 12 and 13.
Thus, the option --playlist-start NUMBER should help you to start the playlist in the middle, specified by NUMBER.