  The one-loop contributions to the decays of the $CP$-odd and $CP$-even scalar
bosons $A\to Z\gamma\gamma$ and $\phi\to Z\gamma\gamma$ ($\phi=h,H$) are
calculated within the framework of $CP$-conserving THDMs, where they are
induced by box and reducible Feynman diagrams. The behavior of the
corresponding branching ratios are then analyzed within the type-II THDM in a
region of the parameter space around the alignment limit and still consistent
with experimental data. It is found that the $A\to Z\gamma\gamma$ branching
ratio is only relevant when $m_A>m_H+m_Z$, but it is negligible otherwise. For
$m_A>600$ GeV and $t_\beta\simeq O(1)$, $BR(A\to Z\gamma\gamma)$ can reach
values of the order of $10^{-5}-10^{-4}$, but it decreases by about one order
of magnitude as $t_\beta$ increases up to 10. A similar behavior is followed by
the $H\to Z\gamma\gamma$ decay, which only has a non-negligible branching ratio
when $m_H>m_A+m_Z$ and can reach the level of $10^{-4}-10^{-3}$ for $m_H>600$
GeV and $t_\beta\simeq O(1)$. We also estimated the branching ratios of these
rare decays in the type-I THDM, where they can be about one order of magnitude
larger than in type-II THDM. As far as the $h\to Z\gamma\gamma$ decay is
concerned, since the properties of this scalar boson must be nearly identical
to those of the SM Higgs boson, the $h\to Z\gamma\gamma$ branching ratio does
not deviates significantly from the SM prediction, where it is negligibly
small, of the order of $10^{-9}$. This result is in agreement with previous
calculations.
