A systematic numerical investigation of the n =1 ( n is the toroidal mode number) internal kink mode (IKM) stability is carried out, for a conventional aspect ratio tokamak plasma in the presence of parallel equilibrium flow or its poloidal/toroidal projections. The computational results, obtained utilizing the recently updated MARS-F code [Y. Q. Liu et al., Phys. Plasmas 7, 3681 (2000)], show that a pure parallel flow provides minor influence on the internal kink instability as well as the mode frequency, being consistent with the intuitive understanding that the parallel flow mainly introduces a rotational transform along the equilibrium magnetic field line. The parallel flow shear somewhat destroys the (uniform) rotational transform, but the eventual (destabilizing) effect on the internal kink is still weak. On the other hand, a much stronger destabilization occurs by keeping only the poloidal or toroidal projection of the parallel flow. The computed mode growth rate is found to be symmetric with respect to the parallel flow direction, whilst the mode frequency is anti-symmetric. These symmetry properties are also confirmed by analytic calculations. The flow shear of the parallel flow component slightly weakens destabilization of the IKM by the poloidal or toroidal projection. The plasma parallel viscosity is found to be strongly stabilizing to the IKM, independent of the parallel flow direction.