Thank you frankz! As a new arrival to Musicbee I might not be making sense, but I have created a library from a folder on my computer containing my audiobooks and named it Audiobook library, and then I created playlists using this library.
Of course, I set all the audiobooks in all the playlists to "remember playback position" but I still get this behaviour. The positions within the tracks are remembered (when the track is clicked directly it is going to resume at the point where I'd stopped, not from the beginning), but the problem is that upon clicking the artwork to start playing the audiobook, the playback always starts from the first track and I don't get any indication which track had been played last.
EDIT: This happens only when selecting albums within a single playlist by clicking on the artwork. When switching between separate playlists by clicking on the playlist in the Playlist explorer, positions are preserved and I can continue listening where I left off.