We present an explicit fourth-order compact finite difference scheme for approximating the three dimensional convection-diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1. We design a parallelization-oriented multigrid method for fast solution of the resulting linear system using a four-color Gauss-Seidel relaxation technique for robustness and efficiency, and a scaled residual injection operator to reduce the cost of multigrid inter-grid transfer operator. Numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method. Several test problems are solved and highly accurate solutions of the 3D convection-diffusion equations are obtained for small to medium values of the grid Reynolds number. Effects of using different residual projection operators are compared on both vector and serial computers.