Parabola: is a two-dimensional, mirror-symmetrical curve, which is
approximately U-shaped when oriented as shown in the diagram below, but
which can be in any orientation in its plane. It fits any of several
superficially different mathematical descriptions which can all be
proved to define curves of exactly the same shape.
Hyperbola: In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type
of smooth curve lying in a plane, defined by its geometric properties or
by equations for which it is the solution set. A hyperbola has two
pieces, called connected components or branches, that are mirror images
of each other and resemble two infinite bows