I've now heard parodies of that song far more times than I've actually ever heard the song itself (which might be twice).

I don't think would call this a parody. It's an interpretation of the original as lounge standard. Despite the strange costume, it's a very well executed cover. Puddles has a lovely voice for this sort of thing and I hope he records more. If you search for him, you find a lot of live stuff which doesn't sound nearly as good as the studio stuff.