  We give an explicit rational parameterization of the surface $\mathcal{H}_3$
over $\mathbb{Q}$ whose points parameterize genus 2 curves $C$ with full
3-level structure on their Jacobians $J$. We use this model to construct
abelian surfaces $A$ such that the Tate-Shafarevich group of $A_d$ has
non-trivial 3-torsion, for a positive proportion of quadratic twists $A_d$. In
fact, for $100\%$ of $x \in \mathcal{H}_3(\mathbb{Q})$, this holds for the
surface $A = \mathrm{Jac}(C_x)/\langle P \rangle$, where $P$ is the marked
point of order 3. Our methods also give an explicit bound on the average rank
of $J_d(\mathbb{Q})$, as well as statistical results on the size of
$\#C_d(\mathbb{Q})$, as $d$ varies through squarefree integers.
