If a parabolic mirror has a light bulb in its focal point, the light rays are reflected at the mirror into parallel light beams, evenly illuminating whatever lies ahead.
In optics, reflections are well studied. The basic question is what happens when parallel light hits a reflective surface.
The case of the parabola is the rare exception. Typically, the reflected light rays will produce another curve of heightened brightness, called its caustic.
For instance, you might have observed a strangely formed curve in a cup of good black tea when horizontal light hits the rim of your circular cup. This curve is actually a nephroid, well studied since antiquity.
Doing the same to an ellipse gives a deformed picture.
Still other curves like the spiral below have elegantly ornamental curves as their caustics.