Modified Nodal Integral Method (MNIM) for three-dimensional, incompressible Navier-Stokes (N-S) equations has recently been developed. A parallel version of the numerical method (PMNIM) has been subsequently developed. Modified Nodal Integral Method (MNIM) requires relatively less number of grid points for the desired accuracy which results in reduced CPU time. The parallel version of the method, PMNIM, is tested for the laminar flow and has been shown to have good speedup and parallel efficiency. The higher efficiency of PMNIM makes it a suitable candidate for simulation of turbulent flows. In the present work, PMNIM is used to for direct numerical simulation of Arnold-Beltrami-Childress (ABC) flow for a range of Reynolds numbers. ABC flows are considered to be prototypes for the study of turbulence. The earlier studies of the chaos in the ABC flows have been carried out using Fourier Spectral Method. The results obtained in previous studies are reproduced in the present work using PMNIM. The results obtained using PMNIM match favourably with those obtained by using Fourier spectral method.