Definitely prog rock but there are some 'jazzy' bits as there often are in prog rock.
I also think it was satire and a parody of 'prog rock' and 'concept' albums, but in the end he ended up creating a fan favorite of 'prog rock'.
An interesting analysis of the lp
https://thegemsbok.com/art-reviews-and-articles/tuesday-tome-thick-brick-jethro-tull-irony/
2005 interview with Anderson:
"In my mind when it came to writing the next album,
Thick as a Brick, was done very much in the sense of: ‘Whuh, if they thought
Aqualung was a concept album, Oh! Okay, we’ll show you a concept album.’ And it was done as a kind of spoof, a send-up, of the concept album genre."
I always felt he should have had 4 long songs on his 'concept' rather than just 2 tracks. I always lose interest about half way through each track.....and I'm a Tull fan btw.