$H^*(Y_n;\mathbb{Q}) \cong (H^*((S^3)^N;\mathbb{Q}))^{\Sigma_n}$.
Hence with rational coefficients this reduces to analyzing the invariant subspace of the action of $\Sigma_n$ on $H^*((S^3)^N;\mathbb{Q})\cong (\mathbb{Q}^2)^{\otimes N}$.
To analyze the invariant subspace, let’s switch to $\mathbb{R}$ coefficients and de Rham cohomology. Let $H^*(S^3;\mathbb{R})$ be generated by a $0$-form the constant function $s^0 \in \Omega^0(S^3) = C^{\infty}(S^3)$ and a volume form $s^3\in \Omega^3(S^3)$. We have a projection $p_i: (S^3)^N \to S^3$ to the $i$th factor. Then $H^*((S^3)^N)$ is generated by $p_1^*(s_{i_1})\wedge \cdots \wedge p_N^*(s_{i_N})$, where $i_j\in \{0,3\}$.