The approximated parabolic mirror works because the second spherical section shares the same axis of symmetry.  Some of its inward directed light rays will pass through the focal point created by the first spherical section. Without the second spherical section, those rays would not have been reflected through the desired focal point.
By adding more spherical sections between the two existing spherical sections, more of the incomming parallel light rays can be diverted through this common focal point. When enough spherical sections are added to reflect all the incomming parallel light rays through the common focal point, the approximated parabolic curve will coincide with the true parabolic curve,- resulting in a true parabolic mirror.
It is the numerous inward directed rays that do not pass through the desired focal point that concern us. Many of these rays that have crossed the axis of symmetry from the inner spherical section, meet similar rays from the outer spherical section before those rays leave their outer strike zone,- forming partial focal points in the outer spherical section's strike zone. This is coma error.