Getting Mario-Mask Koopas in “Super Mario World”

I’ve read about Koopas wearing Mario-masks in Super Mario World, and the pictures below look familiar (though I can’t recall if it’s because I got them in the game or saw them on the Internet).

I can’t find any direct information on how to get them other than “finishing the special world” or “completing 100% of the game”.

Does anyone know exactly what is required of the player to get the masked Koopas and when/where they show up?

Solution 1:

There is a secret warp in the Star Road. Upon entering it, you will end up in the Special World. If you complete all 8 of the Special World levels, a new Star Road appears with a warp that leads back to Yoshi's House. If you enter it, some drastic changes occur in Dinosaur Land:

• World backgrounds change (to a Fall-like instead of a Spring-like colour scheme).
• Koopas transform into Mask Koopas.
• Piranha Plants will change into Pumpkin Plants.
• Jumping Piranha Plants will turn into Jumping Pumpkin Plants.
• Bullet Bills are replaced with Pidgits.
• Goombas turn into a yellowish color and have sunglasses. (GBA version only)
• Pokeys turn into sawblade-like creatures. (GBA version only)

(Source: Super Mario Wiki)

These changes occur in the entire world, and are permanent.

Solution 2:

After beating the final Star Road level, "Funky", a second star-road teleporter will appear to the left of the level. Taking this teleporter will send you all the way back to Yoshi's home back in world 1, and will simultaneously activate the "Special World".

Here's a picture to explain how to avoid triggering the Special World: