How we can differentiate between the shapes of a parabola and a hyperbola? How we can differentiate between the shapes of the conics hyperbolas and parabolas?
Draw two parallel chords of your conic, and let $a$ be the line through the midpoints of the chords. Then draw another pair of parallel chord (not parallel to the previous pair) and let $b$ be the line through their midpoints. If lines $a$ and $b$ are parallel, then the conic is a parabola, otherwise it is a hyperbola (and the lines meet at the center of the hyperbola).

An example can be seen in figure below: on the left a parabola and on the right a branch of hyperbola.
A parabola is the intersection of a plane parallel to the surface of a cone (green) and a hyperbola the plane is parallel to the central axis of the cone (red).