Maybe I misunderstand your point of spherical vs parabolic, but doesn't this paragraph explain it?

> The concept turns part of the inside surface of an inflated sphere into a parabolic antenna. A section comprising about a third of the balloon’s interior surface is aluminized, giving it reflective properties.

The balloon is approximately spherical but probably not exactly (actually, when looking closely, hardly any beachballs are spherical either). What is parabolic is the part of the balloon which is aluminized.

Edit: fixed a typo