The Serre weight conjectures, as extended by Edixhoven, imply a correspondence between unramified odd irreducible mod $p$ Galois representations and weight 1 mod $p$ modular forms. This correspondence has a natural generalization to Hilbert modular varieties, where we now consider the higher mod $p$ coherent cohomology of Hilbert modular varieties. I will talk about a proof of one direction of this correspondence, namely that the Galois representation associated to the mod $p$ coherent cohomology of Hilbert modular varieties in parallel weight 1 is unramified.