  Truncated shifted Yangians are a family of algebras which naturally quantize
slices in the affine Grassmannian. These algebras depend on a choice of two
weights $\lambda$ and $\mu$ for a Lie algebra $\mathfrak{g}$, which we will
assume is simply-laced. In this paper, we relate the category $\mathcal{O}$
over truncated shifted Yangians to categorified tensor products: for a generic
integral choice of parameters, category $\mathcal{O}$ is equivalent to a weight
space in the categorification of a tensor product of fundamental
representations defined by the third author using KLRW algebras. We also give a
precise description of category $\mathcal{O}$ for arbitrary parameters using a
new algebra which we call the parity KLRW algebra. In particular, we confirm
the conjecture of the authors that the highest weights of category
$\mathcal{O}$ are in canonical bijection with a product monomial crystal
depending on the choice of parameters.
  This work also has interesting applications to classical representation
theory. In particular, it allows us to give a classification of simple
Gelfand-Tsetlin modules of $U(\mathfrak{gl}_n)$ and its associated W-algebras.
