The Einstein equations with a cosmological constant, when restricted to Euclidean space-time with anti-self-dual Weyl tensor, can be replaced by a quadratic condition on the curvature of an SU(2) connection. It is known, when the cosmological constant is negative the dimension of the moduli space of gauge-inequivalent solutions to this equation is essentially controlled by the Atiyah-Singer index theorem provided the field equations are linearization stable. It is shown that linearization instability occurs whenever the unperturbed geometry possesses a Killing vector and/or a “harmonic Weyl spinor”. It is then proved that while there are no Killing vectors on compact conformally anti-self-dual Einstein space with a negative cosmological constant, it is possible to have harmonic Weyl spinors. Therefore, the conformally anti-self-dual Einstein equations on a compact Euclidean manifold are linarization stable when the cosmological constant is negative provided the unperturbed geometry admits no harmonic Weyl spinors.