  In this work we have investigated the phenomenological consequences of
two-zero textures of \textit{inverse} neutrino mass matrix ($M_{\nu}^{-1}$) in
light of the large mixing angle (LMA) and large mixing angle-\textit{dark}
(LMA-D) solutions, later of which originates if neutrinos exhibit non-standard
interactions with matter. Out of fifteen possibilities, only seven two-zero
textures of $M_{\nu}^{-1}$ are found to be phenomenologically allowed under LMA
and/or LMA-D descriptions. In particular, five textures are in consonance with
both LMA and LMA-D solutions and are necessarily $CP$ violating while remaining
two textures are found to be consistent with LMA solution only. The textures
with vanishing (1, 1) element of $M_{\nu}^{-1}$ are, in general, disallowed.
All the textures allowed under LMA and LMA-D solutions follow the same neutrino
mass hierarchy. Furthermore, textures with vanishing (2, 3) element of
$M_{\nu}^{-1}$ are found to be either disallowed or are consistent with LMA
description only. We have, also, obtained the implication of the model for
$0\nu\beta\beta$ decay amplitude $|M_{ee}|$. For most of the textures the
calculated $3\sigma$ lower bound on $|M_{ee}|$ is $\mathcal{O}(10^{-2})$, which
is within the sensitivity reach of $0\nu\beta\beta$ decay experiments. We have,
also, proposed a flavor model based on discrete non-Abelian flavor group $A_4$
wherein such textures of $M_{\nu}^{-1}$ can be realized within Type-I seesaw
setting.
