Yep, I've looked at your file and it is the F_SKY1 issue, but obviously I didn't explain it well enough.
When you tell the game which texture to use as the sky via MAPINFO, you are just saying "wherever sky appears, use this texture." You are not saying "wherever I put this texture, treat it like a sky."
F_SKY1 is a special flat that is used as a "sky marker". Whenever it is used on the ceiling (or the floor) of a sector, the game recognises that you have marked that sector as being one that is open to the sky. It then uses the texture you have identified in MAPINFO as the basis for the parallax sky effect.
So, the solution is to put F_SKY1 on the ceiling of any sector that you want to look like sky. e.g. in the case of the map that you uploaded, you used KITTY on the ceiling. I changed it to F_SKY1 and here's the result:
