For unstructured perturbation it has been shown that the robustness of any structurally stable regulating compensator is limited by the minimum distance to a class of "critical" systems. These critical systems also play a central role for parametrized families, but in this case it is not enough merely that such a system exists. Their structure in the parameter space must also be considered. We prove that if the parametrized family contains a critical point, then generically some closed-loop systems in a neighborhood of that point corresponding to any single structurally stable regulating compensator must be exponentially unstable. However, we also show, with a numerical example, that if the family contains no critical points it may be possible to regulate every member - even if a nearby critical point severely limits robust regulation in the unstructured sense. Under these circumstances, the submanifold structure, and in particular the codimension, of the set of critical points in the parameter space becomes a matter of critical interest. We show that in general the critical points form a codimension-two submanifold. However, in special cases of engineering interest, these submanifolds are codimension-one, and hence naturally partition the parameter space.