Parabolic mirrors produce perfect images for infinite-conjugate on-axis points (that is, for perfectly collimated light directed along the optical axis of the parabola).  Here is a diagram of an f/0.3 parabola, shown bringing a collimated beam to perfect focus:

Obviously, parabolic mirrors are a good choice for a collimator in many optical systems, where the parabola is used to produce a collimated beam from a point source.  Using a mirror like the one shown above requires that part of the beam be obscured by the optics in the system that will be used before and/or after the mirror.  In addition, most systems do not require a collimator as fast as f/0.3 – most often a much smaller mirror can be used. 

To get around this obscuration problem, and in the interest of making the mirror as small as possible, an off-axis parabola (OAP) is often used instead of the entire parabola.  An OAP is simply a small section cut out from a full, parent parabola.  Below is an OAP shown in relation to its parent parabola.

Keeping this parent parabola in mind will help when manipulating OAPs.