A moment approach to calculate neoclassical transport in non-axisymmetric torus plasmas composed of multiple ion species is extended to include the external parallel momentum sources due to unbalanced tangential neutral beam injections (NBIs). The momentum sources that are included in the parallel momentum balance are calculated from the collision operators of background particles with fast ions. This method is applied for the clarification of the physical mechanism of the neoclassical parallel ion flows and the multi-ion species effect on them in Heliotron J NBI plasmas. It is found that parallel ion flow can be determined by the balance between the parallel viscosity and the external momentum source in the region where the external source is much larger than the thermodynamic force driven source in the collisional plasmas. This is because the friction between C6+ and D+ prevents a large difference between CC6+ and D+ flow velocities in such plasmas. The C6+ flow velocities, which are measured by the charge exchange recombination spectroscopy system, are numerically evaluated with this method. It is shown that the experimentally measured C6+ impurity flow velocities do not contradict clearly with the neoclassical estimations, and the dependence of parallel flow velocities on the magnetic field ripples is consistent in both results. (C) 2016 AIP Publishing LLC.