  In this paper we consider gravitational parity anomaly in three and four
dimensions. We start with a re-computation of this anomaly on a 3D manifold
without boundaries and with a critical comparison of our results to the
previous calculations. Then we compute the anomaly on 4D manifolds with
boundaries with local bag boundary conditions. We find, that gravitational
parity anomaly is localized on the boundary and contains a gravitational
Chern-Simons terms together with a term depending of the extrinsic curvature.
We also discuss the main properties of the anomaly, as the conformal
invariance, relations between 3D and 4D anomalies, etc.
