For every partition $λ$ of a positive integer $n$, let $S^λ$ be the corresponding Specht module of the symmetric group $\mathfrak{S}_n$, and let $\det(λ)\in \mathbb Z$ denote the Gram determinant of the canonical bilinear form with respect to the standard basis of $S^λ$. Writing $\det(λ)=m \cdot 2^{a_λ^{(2)}}$ for integers $a_λ^{(2)}$ and $m$ with $m$ odd, we show that if the dimension of $S^λ$ is even, then $a_λ^{(2)}$ is also even. This confirms a conjecture posed by Richard Parker in the special case of the symmetric groups.