  Parity violation in QCD is a consequence of the so-called QCD
$\theta$-vacuum. As a result, parity-odd fragmentation functions are introduced
and they bring in new observables in the back-to-back dihadron productions in
$e^+e^-$-annihilation experiments [Phys.Rev.Lett. 106 (2011) 042001]. The
experimental measurements on the corresponding parity-odd fragmentation
functions can shed light on the local CP violation effect in QCD. On the other
hand, the weak interaction also violates the parity symmetry. Therefore, the
weak decay of heavier hadrons can also contribute to the parity-odd effects in
fragmentation functions. In this paper, we investigate the weak decay
contribution to these parity-odd fragmentation functions and compute their
contribution to these new observables. In principle, the decay contribution
should/can be excluded in the theoretical analysis and experimental
measurements. However, this is usually not the common practice so far.
Furthermore, in light of that the value of the $\theta$-parameter is extremely
small ($\theta < 3 \times 10^{-10}$), the weak-parity-violating contributions
become an important background in identifying the strong-parity-violating term.
In this paper, we focus on the weak decay contribution of parity-odd
fragmentation functions and demonstrate their sizable contribution in a
numerical estimate.
