The aim of this paper is to construct regularized asymptotics of the solution of a two-dimensional partial differential equation of parabolic type with a small parameter for all spatial derivatives and a rapidly oscillating free term.      The case when the first derivative of the phase of the free term at the initial point vanishes is considered. The two-dimensionality of the equation leads to the emergence of a two-dimensional boundary layer. The presence in the free term of a rapidly oscillating factor leads to the inclusion in the asymptotic of the boundary layer with a rapidly oscillating nature of change.   The vanishing of the derived phase of the free term introduces into the asymptotic of a new type of boundary layer function. A complete asymptotic solution of the problem is constructed by the method of regularization of singularly perturbed problems developed by S.А. Lomov and adapted by one of the authors for singularly perturbed parabolic equations.