In this article, we consider the setting of single-valued, smoothly varying directions of reflection and non-smooth time-dependent domains whose boundary is Hölder continuous in time. In this setting, we prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. In the same setting, we also prove, using the theory of viscosity solutions, a comparison principle for viscosity solutions to fully nonlinear second-order parabolic partial differential equations and, as a consequence, we obtain existence and uniqueness for this class of equations as well. Our results are generalizations of two articles by Dupuis and Ishii to the setting of time-dependent domains.