Let's see why the study of moduli space of complex structures is closely related to mirror pairs.

From the 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 the BTT Theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for the tangent space of moduli of complex structures.

So from the 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 the 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 a 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 this Kontsevich paper.