Why Parity Anomaly in Odd Dimensions?

Question

In section 13.6 of Nakahara, the parity anomaly is in odd dimensional spacetime. 

From the paper "Fermionic Path Integral And Topological Phases"

https://arxiv.org/abs/1508.04715

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?

I also posted my question here 

https://physics.stackexchange.com/q/436841/185558