  We propose a framework of Finsler space-time to explain the observed parity
asymmetry and the power deficit in the low-$\ell$ ($2\leqslant \ell \leqslant
29$) multipole range of cosmic microwave background (CMB) temperature
anisotropies. In the $3+1$ dimensional space-time, the three-dimensional space
is described by a Randers-Finsler space, which is spatially irreversible,
inducing the parity asymmetry in the CMB angular power spectrum. We estimate
the constraints on the two parameters introduced by Finsler space-time via
analyzing the low-$\ell$ angular power spectrum in PLANCK 2015 CMB temperature
data. We see that the low-$\ell$ power suppression in the CMB temperature
anisotropies can also be resolved in this scenario. Our study shows that the
two low-$\ell$ anomalies, i.e., parity asymmetry and power deficit, may have a
common origin.
