In this paper, we construct a family of temporal high-order parametric relaxation Runge–Kutta (pRRK) schemes for stiff ordinary differential equations (ODEs), and explore their application in solving hyperbolic conservation laws with source terms. The new time discretization methods are explicit, large time-stepping, delay-free and able to preserve steady state. They are combined with fifth-order weighted compact nonlinear schemes (WCNS5) spatial discretization and parametrized maximum-principle-preserving (MPP) flux limiters to solve scalar hyperbolic equations with source terms. We prove that the fully discrete schemes preserve the maximum principle strictly. Through benchmark test problems, we demonstrate that the proposed schemes have fifth-order accuracy in space, fourth-order accuracy in time and allow for large time-stepping without time delay. Both theoretical analyses and numerical experiments are presented to validate the benefits of the proposed schemes.