A new approach called Backward Euler method with Predictor-Corrector (BEM-PC) for simulating cascading failure in dynamic models of power systems was recently proposed. It applied Backward Euler integration method (BEM) with stiff decay property while overcoming its so-called hyperstability issue. The method led to a significant simulation speedup without sacrificing accuracy in tracking cascading path when compared with traditional solution techniques like Trapezoidal integration method (TM). In this paper, we demonstrate that a further speedup can be achieved by a parallelized version of BEM-PC, which we call BEM-PC-parallel (BEM-PCP). In this version, the predictor subprocess of BEM-PC is run in multiple parallel processors for identification of oscillatory instability using eigen-decomposition of the system matrix at post-disturbance unstable equilibria. Monte-Carlo studies on a 2,383-bus Polish system confirm that BEM-PCP is on average 17% faster than BEM-PC and $\approx 40$ times as fast as TM while maintaining the same accuracy as BEM-PC.