  We find the $E$-polynomials of a family of parabolic
$\mathrm{Sp}_{2n}$-character varieties $\mathcal{M}^{\xi}_{n}$ of Riemann
surfaces by constructing a stratification, proving that each stratum has
polynomial count, applying a result of Katz regarding the counting functions,
and finally adding up the resulting $E$-polynomials of the strata. To count the
number of $\mathbb{F}_{q}$-points of the strata, we invoke a formula due to
Frobenius. Our calculation make use of a formula for the evaluation of
characters on semisimple elements coming from Deligne-Lusztig theory, applied
to the character theory of $\mathrm{Sp}{\left(2n,\mathbb{F}_{q}\right)}$, and
M\"obius inversion on the poset of set-partitions. We compute the Euler
characteristic of the $\mathcal{M}^{\xi}_{n}$ with these polynomials, and show
they are connected.
