Note that $H^{0,0}_{\bar{\partial}}(X) = \ker\bar{\partial} : A^{0, 0}(X) \to A^{0, 1}(X)$ is precisely the collection of holomorphic functions on $X$ which, historically, were of more interest than anti-holomorphic functions on $X$ which is given by $H^{0,0}_{\partial}(X) = \ker\partial : A^{0,0}(X) \to A^{1,0}(X)$.
Moreover, Dolbeault cohomology $H^{p,q}_{\bar{\partial}}(X)$ determines $H^{p,q}_{\partial}(X)$ and vice versa. More precisely, the maps
\begin{align*}
\psi : H^{p,q}_{\bar{\partial}}(X) &\to H^{q,p}_{\partial}(X)\\
[\alpha] &\mapsto [\bar{\alpha}]
\end{align*}
and
\begin{align*}
\varphi : H^{q,p}_{\partial}(X) &\to H^{p,q}_{\bar{\partial}}(X)\\
[\beta] &\mapsto [\bar{\beta}]
\end{align*}
are well-defined and are inverses of one another. Therefore $H^{p,q}_{\bar{\partial}}(X) \cong H^{q,p}_{\partial}(X)$.
If $X$ is compact, then Hodge Theory tells us that $H^{p,q}_{\bar{\partial}}(X) \cong \ker\Delta_{\bar{\partial}} : A^{p,q}(X) \to A^{p,q}(X)$ and $H^{p,q}_{\partial}(X) \cong \ker\Delta_{\partial} : A^{p,q}(X) \to A^{p,q}(X)$ where $\Delta_{\bar{\partial}}$ and $\Delta_{\partial}$ are the holomorphic and anti-holomorphic Laplacians respectively. If $X$ is also Kähler, then $\Delta_{\bar{\partial}} = \Delta_{\partial}$ so we see that
\begin{align*}
H^{p,q}_{\bar{\partial}}(X) &\cong \ker\Delta_{\bar{\partial}} : A^{p,q}(X) \to A^{p,q}(X)\\
&= \ker\Delta_{\partial} : A^{p,q}(X) \to A^{p,q}(X)\\
&\cong H^{p,q}_{\partial}(X).
\end{align*}
Combining these isomorphisms with the ones above, we see that $H^{p,q}_{\bar{\partial}}(X) \cong H^{q,p}_{\bar{\partial}}(X)$ so the Hodge numbers satisfy $h^{p,q} = h^{q,p}$. Therefore the odd Betti numbers of a compact Kähler manifold are even.
