A new parallel R-point explicit block method for solving second order ordinary differential equations (ODEs) directly is developed. The method computes the numerical solution of the equations at three points simultaneously. Each problem was tested on the shared memory parallel computer Sequent S27 using both the sequential and parallel implementations of the new method and the conventional 1-point method. Numerical results are presented comparing the two methods in terms of the number of steps taken, accuracy and execution times. The results indicate the advantages of using the parallel implementation of the new method for solving second order ODEs directly particularly when the step size becomes finer.