Highest weight categories arising from Khovanov's diagram algebra III: category $\mathcal{O}$

We prove that integral blocks of parabolic category $\mathcal{O}$ associated to the subalgebra $\mathfrak{gl}_m(\mathbb{C}) \oplus \mathfrak{gl}_n(\mathbb{C})$ of $\mathfrak{gl}_{m+n}(\mathbb{C})$ are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of $2$-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type $A$.