'Para' means “at or to one side of, beside, side by side” in Greek. So, paraxial rays are rays that are very close to the principal axis. Ergo, the paraxial approximation means that all the rays falling on the mirror are paraxial. Only then will the spherical mirror act as a focusing mirror, as only then will the small spherical portion be close to a parabolic mirror (which is THE focusing mirror!), as shown below.
Note that paraxial rays imply that all the angles involved are really small, which shall allow us to make various approximations (read simplifications) in the coming modules.