For god's sake.
It is the FIRST in a SERIES OF MUSIC VIDEOS using the SOUNDTRACK OF THE TV SERIES. It, and ANY OTHER future works in the series, are NOT trailers for future Gurren-Lagann projects.
It is called PARALLEL WORKS because it is in a PARALLEL universe where only the CHARACTERS APPEARANCES AND PERSONALITIES and such are the same. Not all of them will be medieval, they can be whatever each independant director wants them to be.
If you want it to be canon, you could possibly consider each video as one of the alternate universes within the Infinite Space Labyrinth in episode 26.
Also, that's not Lazengann. It's the Grand Zamboa (the Anti-Spiral's mech). Look at the eyes and spike positions, as well as facial features.