  In this paper we provide new rigidity results for four-dimensional Riemannian
manifolds and their twistor spaces. In particular, using the moving frame
method, we prove that $\mathbb{CP}^3$ is the only twistor space whose Bochner
tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally
symmetric twistor spaces and we show the nonexistence of conformally flat
twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer
concerning the self-duality of a Riemannian four-manifold.
