We generalize a result of Lue Pan on locally analytic completed cohomology of modular curves to the case of Hilbert modular varieties. As an application, we prove that for parallel weight Hecke classes appearing in the completed cohomology of Hilbert modular varieties, de Rhamness of the associated Galois representation will imply classicality. We need to understand certain “partial de Rham cohomology”, which we will prove to be classical by developing a version of Jacquet-Langlands transfer of D-modules. We will make use of the plectic Lie algebra action in the joint work with Lue Pan.