This paper considers the mixed covolume method for the second-order elliptic equations over quadrilaterals. Superconvergence results are established in this paper on quadrilateral grids satisfying the h2-parallelogram condition when the lowest-order Raviart-Thomas space is employed in the mixed covolume method. The authors prove O(h2) accuracy between the approximate velocity or pressure and a suitable projection of the real velocity or pressure in the L2 norm. Numerical experiments illustrating the theoretical results are provided.