We present a new two-stage optical design for parabolic dish concentrators that can realistically attain close to 90% of the thermodynamic limit to concentration with practical, compact designs (e.g., at parabola rim half-angles of around 45°). For comparison, the parabolic dish-plus-compound parabolic concentrator secondary design, at this rim angle, achieves no more than 50% of the thermodynamic limit. Our new secondary concentrator is tailored to accept edge rays from the parabolic primary, and incurs less than one reflection on average. It necessitates displacing the absorber from the parabola's focal plane, along the concentrator's optic axis, toward the primary reflector, and constructing the secondary between the absorber and the primary. The secondary tailored edge-ray concentrators described here create new possibilities for building compact, extremely high flux solar furnaces and/or commercial parabolic dish systems.