Recent studies have revealed that viscous coupling effects in immiscible two-phase flow, caused by momentum transfer between the two fluid phases, can be important in porous medium systems. In this work, we use a three-dimensional parallel processing version of a two-fluid-phase lattice Boltzmann (LB) model to investigate this phenomenon. A multiple-relaxation-time (MRT) approximation of the LB equations is used in the simulator, which leads to a viscosity-independent velocity field. We validate our model by verifying the velocity profile for two-phase flow through a channel with a square cross section. We then simulate co-current flow through a sphere-pack porous medium and obtain correlations of the relative permeabilities as a function of capillary number, wettability, and the fluid viscosities. The results are qualitatively consistent with experimental observations. In addition, we calculate the generalized permeability coefficients and show that the coupling coefficients are significant and the matrix is nonsymmetric. We also find a strong correlation between the relative permeability and interfacial area between fluids, indicating that both the common extension of Darcy's Law and the generalized formulation accounting for viscous coupling effects do not provide adequate insight into two-phase flow processes in porous media. This work lends additional support for the hypothesis that interfacial area is a key variable for multiphase flow in porous medium systems.