  Recently, a class of Dirac semimetals, such as \textrm{Na}$_{\mathrm{3}}%
$\textrm{Bi} and \textrm{Cd}$_{\mathrm{2}}$\textrm{As}$_{\mathrm{3}}$, are
discovered to carry $\mathbb{Z}_{2}$ monopole charges. We present an
experimental mechanism to realize the $\mathbb{Z}_{2}$ anomaly in regard to the
$\mathbb{Z}_{2}$ topological charges, and propose to probe it by
magnetotransport measurement. In analogy to the chiral anomaly in a Weyl
semimetal, the acceleration of electrons by a spin bias along the magnetic
field can create a $\mathbb{Z}_{2}$ charge imbalance between the Dirac points,
the relaxation of which contributes a measurable positive longitudinal spin
magnetoconductivity (LSMC) to the system. The $\mathbb{Z}_{2}$ anomaly induced
LSMC is a spin version of the longitudinal magnetoconductivity (LMC) due to the
chiral anomaly, which possesses all characters of the chiral anomaly induced
LMC. While the chiral anomaly in the topological Dirac semimetal is very
sensitive to local magnetic impurities, the $\mathbb{Z}_{2}$ anomaly is found
to be immune to local magnetic disorder. It is further demonstrated that the
quadratic or linear field dependence of the positive LMC is not unique to the
chiral anomaly. Base on this, we argue that the periodic-in-$1/B$ quantum
oscillations superposed on the positive LSMC can serve as a fingerprint of the
$\mathbb{Z}_{2}$ anomaly in topological Dirac semimetals.
