The propagation of optical pulses inside dispersion-managed fibers is considered. It is found that the chirped compact parabolic pulse can propagate inside the dispersion-managed fibers self-similarly. Such a finite-width pulse can be served as the background for the propagation and interaction of dark similaritons. Approximate but highly accurate analytical methods are proposed to describe the interaction dynamics of multiple dark similaritons on the self-similar compact parabolic background.