  Recent developments in solid state physics give a prospect to observe the
parity anomaly in (2+1)D massive Dirac systems. Here we show, that the quantum
anomalous Hall (QAH) state in orbital magnetic fields originates from the Dirac
mass term and induces an anomalous four-current related to the parity anomaly.
This differentiates the QAH from the quantum Hall (QH) state for the
experimentally relevant case of an effective constant density (seen by the
gate). A direct signature of QAH phase in magnetic fields is a long
$\sigma_{xy}= e^2/h$ ($\sigma_{xy}= -e^2/h$) plateau in
Cr$_x$(Bi$_{1-y}$Sb$_y$)$_{2-x}$Te$_3$ (HgMnTe quantum wells). Furthermore, we
predict a new transition between the quantum spin Hall (QSH) and the QAH state
in magnetic fields, for constant effective carrier density, without magnetic
impurities but driven by effective g-factors and particle-hole asymmetry. This
transition can be related to the stability of edge states in the Dirac mass gap
of 2D topological insulators (TIs), even in high magnetic fields.
