In this paper we are interested in solving parabolic problems with a piecewise constant diffusion coefficient on structured Cartesian meshes. The aim of this paper is to investigate the applicability and convergence behavior of combining two non-conventional but innovative methods: the Laplace transformation method in the discretization of the time variable and the immerse finite element method (IFEM) in the discretization of the space variable. The Laplace transformation in time leads to a set of Helmholtz-like problems independent of each other, which can be solved in highly parallel. The employment of immerse finite elements (IFEs) makes it possible to use a structured mesh, such as a simple Cartesian mesh, for the discretization of the space variable even if the material interface (across which the diffusion coefficient is discontinuous) is non-trivial. Numerical examples presented indicate that the combination of these two methods can perform optimally from the point of view of the degrees of polynomial spaces employed in the IFE spaces.