For a positive integer $n$, a partition of $n$ is a sequence $\lambda = (\lambda_1, \dots, \lambda_l)$ of integers with $\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_l \geq 1$ and $ \sum_{i=1}^{l}\lambda_i=n$. Let $R = K[x_1, \dots, x_n]$ be the polynomial ring over a field $K$. For a given partition $\lambda$ of $n$ let $T $ be a Young tableau. If the $j$-th column of $T$ is represented by $j_1,j_2, \dots , j_m$ in top-to-bottom order, one defines $f_T(j)=\prod_{1\leq s < t \leq m}(x_{j_s}-x_{j_t}) \in R$. The Specht polynomial $f_T$ associated with $T$ is given by $f_T=\prod_{j=1}^{\lambda_1}f_T(j)$. For a partition $\lambda$ of $n \in\Bbb{ N}$, let $ I_{\lambda}^{S_p}$ be the ideal of $R = K[x_1, \dots, x_n]$ generated by all Specht polynomials of shape $\lambda$. Assuming ${\rm char}(K)=0$, it has been established that $ R/I_{(n-2, 2)}^{S_p}$ is a Gorenstein ring, and $ R/I_{(d,d, 1)}^{S_p}$ is a Cohen-Macaulay ring with a linear free resolution. In this study, the authors present minimal free resolutions for these rings. C. Berkesch, S. Griffeth and S.~V. Sam [Comm. Math. Phys. {\bf 330} (2014), no.~1, 415--434; MR3215587] had previously investigated minimal free resolutions of $ R/I_{(n-d, d)}^{S_p}$, which is also Cohen-Macaulay, utilizing advanced representation theory techniques. However, the authors' approach relies solely on the fundamental theory of Specht modules, offering explicit descriptions of the differential maps.