  In this paper, we complete the analysis initiated in [AFV24] establishing
some higher order $C^{k+2,\alpha}$ Schauder estimates ($k \in \mathbb{N}$) for
a a class of parabolic equations with weights that are degenerate/singular on a
characteristic hyperplane. The $C^{2,\alpha}$-estimates are obtained through a
blow-up argument and a Liouville theorem, while the higher order estimates are
obtained by a fine iteration procedure. As a byproduct, we present two
applications. First, we prove similar Schauder estimates when the
degeneracy/singularity of the weight occurs on a regular hypersurface of
cylindrical type. Second, we provide an alternative proof of the higher order
boundary Harnack principles established in [BG16,Kuk22].
