A theoretical model shows that in the context of a Ginzburg-Landau equation with rapidly varying, mean-zero dispersion, stable and attracting self-similar breathers are formed with parabolic profiles. These self-similar solutions are the final solution state of the system, not a long-time, intermediate asymptotic behavior. A transformation shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly varying, mean-zero diffusion coefficient. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the parabolic profiles which are, to leading order, of the Barenblatt form. Our analytic model proposes a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model.