The maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations. To propose MBP-preserving schemes with high spatial accuracy, in the first part of this series, we developed a class of time semidiscrete stochastic Runge-Kutta (SRK) methods for semilinear parabolic equations, and constructed the first- and second-order fully discrete MBP-preserving SRK schemes. In this paper, to develop higher order fully discrete MBP-preserving SRK schemes with spectral accuracy in space, we use the Sinc quadrature rule to approximate the conditional expectations in the time semidiscrete SRK methods and propose a class of fully discrete MBP-preserving SRK schemes with up to fourth-order accuracy in time for semilinear equations. Based on the property of the Sinc quadrature rule, we theoretically prove that the proposed fully discrete SRK schemes preserve the MBP and can achieve an exponential order convergence rate in space. In addition, we reveal that the conditional expectation with respect to the Bronwian motion in the time semidiscrete SRK method is essentially equivalent to the exponential Laplacian operator under the periodic boundary condition. Ample numerical experiments are also performed to demonstrate our theoretical results and to show the exponential order convergence rate in space of the proposed schemes.