One way to think about this picture is that the blocks (EDIT: infinitesimal blocks) of the full category $\mathcal{O}$ in this picture aren't just equivalent; they are equivalent in a way that preserves the labeling of simple modules by elements of the Weyl group. So, any module that has some nice characterization using these labels and categorical properties must be preserved. So, the projective cover $P_{w\xi}$ of a simple $L_{w\xi}$ must be sent to $P_{w\nu}$.
Similarly, the Verma module $M_{w\xi}$ must be sent to $M_{w\nu}$ since it is largest quotient of $P_{w\xi}$ where $L_{w'\xi}$ for $w'<w$ in Bruhat order is not a composition factor.
You can describe a parabolic Verma module in a similar way; being in parabolic category $\mathcal{O}$ is just a question about what your composition factors are; you can only have $L_{w\xi}$ for $w$ shortest in a left coset of $W_{I}$ and longest in a right coset for the stabilizer of $\xi$. We can characterize $N_{w\xi}$ as the maximal quotient of $M_{w\xi}$ which only has these composition factors, and thus it must be sent to $N_{w\nu}$.