  We give an explicit rational parameterization of the surface $\mathcal{H}_3$
over $\mathbb{Q}$ whose points parameterize genus 2 curves~$C$ with full
$\sqrt{3}$-level structure on their Jacobian $J$. We use this model to
construct abelian surfaces $A$ with the property that $\mathrm{Sha}(A_d)[3]
\neq 0$ 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.
