Actually this lot are so good, I don’t think of them as a ‘joke’ band at all.
I believe they started out doing U2 covers, hence the name and originally it started out as a bit of a laugh, however they turned out to be pretty darned good
Steve Martin used to make the banjo a big part of his standup routine - and while he never tried too hard to sing (in fact he was terrible), he’s an amazing banjo player.
They Might Be Giants are, on the face of it, a very unserious band with quite unconventional sounding music. They have actually made some very good songs with lyrics on much more serious topics than they can seem as well as some great imagery.
Excellent choice. I think A Day in the Life is the standout on that album.
I heard them a little bit back in the '80s. Just in the last few months I went looking to see if there were any recording or info online, and didn’t find much. Don’t know why I didn’t turn up that site.
Frank Zappa and the Mothers of Invention did very jokey stuff, yet musically ambitious, right from their earliest days until the end.
From Mexico, there is Cafe Tacuba. Their magnu opus, an album called “Re”, included songs from about a dozen genres, each competently composed and played. Most of it is more “hommages” than “jokes”, though.
ETA: Neither Zappa nor Cafe Tacuba would be considered a “joke band” (well, some might call Zappa that), so not quite in the spirit of the OP, but close.
Ween is in a similar vein. If not for their incessant juvenility and potty humor, a lot more of their songs would be on the “homage” side of a “joke song”.
Like Weird Al, a lot of Ween’s musical talent is the ability to spot-on mimic any style.
Although you could argue that they’re not really a “joke” band anymore. Over the years they’ve sort-of evolved from “joke band” to a real heavy metal band that sings about jokey stuff.
You could also probably argue about how musically talented they actually are. I like 'em though. Gwar gwar gwar gwar gwar gwar!