Lets see why the study of moduli space of complex structures is closely related to mirror pair.
From mirror map, we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$
where $M$ and $M^\vee$ are mirror to each other.
We have the following relation for tangent space of moduli space of symplectic structures
$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$
and from BTT Theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for tangent space of moduli of complex structures.
So from mirror map , we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$
More generally if we take extended moduli space of complex and symplectic structures then
$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$
and
$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$
and hence from extended mirror map we can identify
$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself Calabi-Yau variety then
$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$
See Kontsevich paper