In order to improve the low output accuracy caused by the elastic deformations of the branch chains, a finite element-based dynamic accuracy analysis method for parallel mechanisms is proposed in this paper. First, taking a 5–prismatic–spherical–spherical (PSS)/universal–prismatic–universal (UPU) parallel mechanism as an example, the error model is established by a closed vector chain method while its influence on the dynamic accuracy of the parallel mechanism is analyzed through numerical calculation and simulation. According to the structural and error characteristics of the parallel mechanism, a vector calibration algorithm is proposed to reduce the position and pose errors along the whole motion trajectory. Then, considering the elastic deformation of the rod, the rigid-flexible coupling dynamic equations of the mechanism are established by combining the finite element method with the Lagrange method, and the equations are vectorially superimposed by means of internal force cancellation to synthesize the elastic dynamics equation of the connecting rod. Based on the constraint condition of each moving part, the elastodynamic model of the whole machine is obtained. Furthermore, the effect of component flexibility on the dimensionless root mean square error of the displacement, velocity and acceleration of the moving platform is investigated by using a Newmark method, and the dynamic accuracy influenced by these dimensionless root mean square errors is further studied. The research work establishes an important theoretical foundation for the development of the prototype.