The authors describe some results of the project constructing the extrinsic dif\-fe\-ren\-ti\-al geometry on submanifolds of Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). The authors review classification results on totally umbilic space-like hypersurfaces in pseudo-spheres. Such totally umbilic space-like hypersurfaces are considered as model hypersurfaces. In a previous paper, \textit{S. Izumiya} [Legendrian dualities and space-like hypersurfaces in the lightcone, Preprint], a basic Legendrian duality theorem between pseudo-spheres in Minkowski space is shown. In the present paper, space-like parallels and caustics are defined as an application of the basic Legendrian duality theorem. By definition, it is shown that the caustics is the locus of singularities of space-like parallels. Ac\-co\-r\-ding to the classification results of the totally umbilic space-like hypersurfaces in pseudo-spheres, the evolutes are defined. In order to study parallels and evolutes, time-like height functions and space-like height functions are introduced. By direct calculation, the authors show that the above notions of caustics and evolutes are the same. Parallels and caustics are investigated from the viewpoint of Lagrangian or Legendrian singularity theory. The geometric meaning of both the singularities of parallels and evolutes from the viewpoint of the contact with families of model hypersurfaces (totally umbilic hypersurfaces) is studied. Generic properties are investigated. The authors apply the classification results to the case \(n = 3\) and draw some pictures.