This paper deals with predictor-corrector (PC) iteration of Runge-Kutta-Nyström (RKN) methods for integrating initial-value problems for special second-order, ordinary differential equations. We consider RKN correctors based on direct collocation techniques with high stage-order so that the stage-values can be used by extrapolation techniques to construct high-order predictior methods with no additional sequential right-hand side evaluations and without increasing number of processors. Having the high-order predictor in addition with the small convergence factor and error constant, the parallel-iterated PC methods based on direct collocation RKN correctors considered in this paper show the improved efficiency when they are compared to the PIRKN methods available in the literature.