Schauder type estimates for degenerate or singular parabolic systems with partially DMO coefficients

We study elliptic and parabolic systems in divergence form with degenerate or singular coefficients. Under the conormal boundary condition on the flat boundary, we establish boundary Schauder type estimates when the coefficients have partially Dini mean oscillation. Moreover, as an application, we achieve $k^{\text{th}}$ higher-order boundary Harnack principles for uniformly parabolic equations with Hölder coefficients, extending a recent result in [Audrito-Fioravanti-Vita 25] from $k\ge 2$ to any $k\ge 1$.