Except the surface of a beachball is spherical, not parabolic. A parabolic surface is not closed, so you can't inflate it like a beachball.
So I'd like to know either: how they control the shape of the surface, so the mirrored portion is parabolic; or whether the antenna surface is actually spherical, and not parabolic at all.
The portion of a parabolic mirror nearest the focus approximates to a sphere. If the deviation is less than 1/4 wavelength of the signal of interest, a spherical mirror will focus the signal perfectly, as if it were a paraboloid.
A paraboloid is what you need for distortion-free imaging; this antenna is apparently used only for signalling, so perhaps accuracy doesn't matter, because only the gain is important. But this is a NASA publication, so I wouldn't expect them to say it's parabolic if it's really spherical.