In section 13.6 of Nakahara, the parity anomaly is in odd dimensional spacetime.
From the paper Fermionic Path Integral And Topological Phases by Witten, the problem appears as one cannot define the sign of the path integral,
$$S[\bar{\psi},\psi;A]=\int d^{2n+1}x\bar{\psi}iD \!\!\!\!/\,\psi,$$ $$\mathcal{Z}=\det(iD \!\!\!\!/\,)=\prod_{\lambda\in\mathrm{spec}}\lambda,$$
because there are infinite number of positive and negative eigenvalues $\lambda$.
The number of eigenvalues flowing through $\lambda=0$ is related with the index theorem in $2n+2$ dimenions.
Does the partiy anomaly appear in even dimensions?
From Nakahara's derivation, I don't see anything related with the dimension of spacetime. If this anomaly exists in odd dimensions, then why doesn't it appear in even dimensions?
