On the Gram determinants of the Specht modules

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