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Amanita Muscaria is the name of those iconic red toadstools with white spots you see everywhere in pop culture. They've become the generic quintessential mushroom, hence why Parasect has them. They're just what everyone thinks of when they think of mushrooms.
Ingesting them gives people a psychoactive experience of feeling either much larger (Mario mushrooms) or smaller (old fairy paintings,) than they really are. This concept was popularized by the size-altering mushrooms in Alice in Wonderland.
So they're red and white, and round, and wouldn't you know it so are Pokéballs. So the concept of Foongus and Amoongus really isn't that random. Further, Laverre City is full of red and white mushrooms because of the aforementioned association with fairies. Where do you find Foongus in X and Y? On the route outside Laverre. What ELSE is outside Laverre? The Kalos Pokéball factory.