We present a mixed covolume method for a system of first order partial differential equations resulting from the mixed formulation of the general self-adjoint parabolic problem with a variable nondiagonal diffusion tensor. The lowest order Raviart-Thomas mixed element space on rectangles is used. We prove the first order optimal rate of convergence for approximate pressure as well as for approximate velocity. We also prove the second order superconvergence both for approximate velocity and pressure in certain discrete norms.