Yes, there are QFTs with parity-invariant actions which have a parity anomaly. This means that there is no regularization of the path integral which is compatible with parity symmetry. Consequently, any path integral measure constructed by taking the continuum limit of these regularizations will not be parity invariant.

Parity anomalies occur in some 2+1 dimensional gauge theories, when the gauge field is coupled to an odd number of Majorana fermions and the gauge group has odd dual Coxeter number.

The [wikipedia][1] page actually has a decent explanation.
[1]: https://en.wikipedia.org/wiki/Parity_anomaly

Yes, there are QFTs with parity-invariant actions which have a parity anomaly. This means that there is no regularization of the path integral which is compatible with parity symmetry. Consequently, any path integral measure constructed by taking the continuum limit of these regularizations will not be parity invariant.

Parity anomalies occur in some 2+1 dimensional gauge theories, when the gauge field is coupled to an odd number of Majorana fermions and the gauge group has odd dual Coxeter number.

The wikipedia page actually has a decent explanation.