To explore the influence of viscoelastic joint on the propagation law of stress waves in rock mass, the Poyting-Thomson model was introduced as the discontinuous displacement condition, and based on the time-domain recursive method, the propagation equation of the stress waves through a set of parallel viscoelastic joints was derived. Then, the time-domain recursive numerical solutions obtained by degrading the Poyting-Thomson model to the Maxwell model and the Kelvin model were compared with the existing closed-form solutions in the frequency domain to verify the validity of the derivation process. Finally, the influences of relevant parameters were further analyzed. Results show that the non-dimensional coefficients, joint thickness, joint spacing and incident angle of the model all affected the wave propagation. For a single joint, the transmission and reflection coefficient along with the energy dissipation rate of the stress waves mainly depended on the non-dimensional parameters, the incident angle, and the non-dimensional joint thickness of the displacement discontinuity model. For a group of parallel joints, the transmission coefficient was also related to the number of parallel joints and the size of the joint spacing. Besides, the increase in the number of joints had an obvious attenuation effect on the amplitude of the transmitted wave at the critical incident angle.