This paper investigates a class of explicit pseudo three-step Runge-Kuttamethods for arbitrarily high order nonstiff initial value problems for systemsof first-order differential equations. By using collocation techniques and bysuitably choosing collocation points we can obtain a stable s-stage explicit pseudothree-step Runge-Kutta method (EPThRK method) of order p = 2s requiring onlyone effective sequential f- evaluation per step on s-processor parallel computers.By a few widely-used test problems, we show the superiority of the new EPThRKmethods proposed in this paper over red well-known parallel PIRK codes andefficient sequential ODEX, DOPRI5 and DOP853 codes available in the literature.This paper investigates a class of explicit pseudo three-step Runge-Kuttamethods for arbitrarily high order nonstiff initial value problems for systemsof first-order differential equations. By using collocation techniques and bysuitably choosing collocation points we can obtain a stable s-stage explicit pseudothree-step Runge-Kutta method (EPThRK method) of order p = 2s requiring onlyone effective sequential f- evaluation per step on s-processor parallel computers.By a few widely-used test problems, we show the superiority of the new EPThRKmethods proposed in this paper over red well-known parallel PIRK codes andefficient sequential ODEX, DOPRI5 and DOP853 codes available in the literature.