Brown et al. provide a representation of a spread of the Tits quadrangle $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,q)$, $q$ even, in terms of a certain family of $q$ ovals of $\mathrm PG(2,q)$. By combining this representation with the Vandendriessche classification of hyperovals in $\mathrm PG(2,64)$ and the classification of flocks of the quadratic cone in $\mathrm PG(3,64)$, recently given by the authors, in this paper, we classify all the spreads of $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,64)$, up to equivalence. These complete the classification of spreads of $T_2(\mathcal O)$ for $q\le 64$.