We consider the stability of a class of parameterized conic programming problems which are more general than the C2-smooth parameterization. We show that when the Karush-Kuhn-Tucker (KKT) condition, the constraint nondegeneracy condition, the strict complementary condition and the second order sucient condition (named as Jacobian uniqueness conditions here) are satis ed at a feasible point of the original problem, the Jacobian uniqueness conditions of the perturbed problem also hold at some feasible point.