  We analyse the parity-odd ("type P") surface anomalies of the energy-momentum
tensor correlators in conformal field theories, with an emphasis on d=4 and d=3
dimensional spacetimes. Using cohomology analysis we construct the expression
for the most general P-type surface trace anomaly on a singular 2-dimensional
surface in 4-dimensional bulk spacetimes. As an important example, we
specialise to the case when the singular surface is a conical defect and show
that the bulk P-type Pontryagin trace anomaly induces such a surface trace
anomaly. We show that this conical type P surface trace anomaly is given purely
by the outer curvature tensor. In addition, we analyse parity-odd surface
contact terms in energy-momentum tensor correlators in the flat spacetime
induced by the conical defect by studying two special cases in which the
contact terms are induced by, (1) type P trace anomaly in d=4 and, (2)
gravitational Chern-Simons Lagrangian term in d=3 spacetime dimensions. In both
cases we show that the surface contact terms appear in correlators of the lower
rank than the corresponding bulk surface terms.
