As already pointed out, the Hodge numbers may go up under reduction modulo $p$. On the other hand, let me also point out that the situation can be controlled:
1.) For all $p$, where $\overline{X}_p$ is smooth, the $\ell$-adic Betti numbers of $X$ and $\overline{X}_p$ are the same.
2.) Now, by the universal coefficient formula relating crystalline and deRham cohomology, we have for all $i$ short exact sequences
$$ 0 \to H^i_{cris}(\overline{X}_p/W)/p\to $$
$$H^i_{dR}(\overline{X}_p/k_p)\to {\rm Tor}_1^{W(k_p)}(H_{cris}^{i+1}(\overline{X}_p/W),k_p)\to 0$$
where $k_p={\cal O}_K/p$. Now, if $\overline{X}_p$ has torsion-free crystalline cohomology, then the term on the right is zero, and the term on the left is a $k_p$-vector space of dimension equal to the $i$.th $\ell$-adic Betti number.Then, the Fr\"olicher spectral sequence relating Hodge- and deRham-cohomology degenerates at $E_2$, we have that $\sum_{p+q=i}h^{p,q}$ is equal to the $i$.th $\ell$-adic Betti number. Thus, simply for dimension reasons, the Hodge numbers of $X$ and $\overline{X}_p$.
The upshot is that torsion in crystalline cohomology of $\overline{X}_p$ detects and controls the differences in Hodge numbers of $X$ and $\overline{X}_p$. For almost all $p\in {\rm Spec} {\cal O}_K$, the reduction $\overline{X}_p$ will be smooth and will have torsion-free crystalline cohomology.