  Recent experimental progress in condensed matter physics enables the
observation of signatures of the parity anomaly in two-dimensional Dirac-like
materials. Using effective field theories and analyzing band structures in
external out-of-plane magnetic fields (orbital fields), we show that
topological properties of quantum anomalous Hall (QAH) insulators are related
to the parity anomaly. We demonstrate that the QAH phase survives in orbital
fields, violates the Onsager relation, and can be therefore distinguished from
a quantum Hall (QH) phase. As a fingerprint of the QAH phase in increasing
orbital fields, we predict a transition from a quantized Hall plateau with
$\sigma_\mathrm{xy}= -\mathrm{e}^2/\mathrm{h}$ to a not perfectly quantized
plateau, caused by scattering processes between counterpropagating QH and QAH
edge states. This transition can be especially important in paramagnetic QAH
insulators, such as (Hg,Mn)Te/CdTe quantum wells, in which exchange interaction
and orbital fields compete.
