$H^*(Y_n;\mathbb{Q}) \cong (H^*((S^3)^N;\mathbb{Q}))^{\Sigma_n}$.
Hence with rational coefficients this reduces to analyzing the invariant subspace of the action of $\Sigma_n$ on $H^*((S^3)^N;\mathbb{Q})\cong (\mathbb{Q}^2)^{\otimes N}$.
I suspect that this will correspond to the number of simple graphs with $n$ vertices. https://oeis.org/A000088 Each such graph gives a choice of cohomology class in $H^*((S^3)^N;\mathbb{Q})$, and hence in $H^*(Y_n)$ by averaging over the action of $\Sigma_n$. I suspect that these classes are linearly independent, but I haven’t checked it.