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.
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