  We analyze the parity-odd correlators $\langle JJO\rangle_{odd}$, $\langle
JJT\rangle_{odd}$, $\langle TTO\rangle_{odd}$ and $\langle TTT\rangle_{odd}$ in
momentum space, constrained by conformal Ward identities, extending our former
investigation of the parity-odd chiral anomaly vertex. We investigate how the
presence of parity-odd trace anomalies affect such correlators. Motivations for
this study come from holography, early universe cosmology and from a recent
debate on the chiral trace anomaly of a Weyl fermion. In the current CFT
analysis, $O$ can be either a scalar or a pseudoscalar operator and it can be
identified with the trace of the stress energy tensor. We find that the
$\langle JJO\rangle_{odd}$ and $\langle TTO\rangle_{odd}$ can be different from
zero in a CFT. This occurs when the conformal dimension of the scalar operator
is $\Delta_3=4$, as in the case of $O=T^\mu_\mu$. Moreover, if we assume the
existence of parity-odd trace anomalies, the conformal $\langle
JJT\rangle_{odd}$ and $\langle TTT\rangle_{odd}$ are nonzero. In particular, in
the case of $\langle JJT\rangle_{odd}$ the transverse-traceless component is
constrained to vanish, and the correlator is determined only by the trace part
with the anomaly pole.
