In this research, using an approximate analytical method, vibration analysis of a 3-PRP (active prismatic—P, passive revolute—R, passive prismatic—P) planar parallel robot having a flexible moving platform is presented. A specific architecture of the 3-PRP parallel robot, also known as the ST (Star-Triangle) parallel robot, is considered. The moving platform of the robot, called the star, is assumed to be made of three flexible beams shaped like a star. For analytical modeling, each of the three beams of the star is assumed to be a discrete Euler–Bernoulli beam with a passive prismatic joint. Continuity equations at the center of the star are used to relate vibrations of the three beams. The vibration behavior of each beam is modeled using previously developed constrained motion equations for a planar Euler–Bernoulli beam having a prismatic joint. In this paper, previously presented “constrained assumed modes method” is further developed to solve the constrained motion equation for the ST parallel robot. The solution method is used to obtain the vibration of the robot for the inverse dynamics problem and simultaneously provides generalized constraint forces. Furthermore, the solution method can be used for the direct dynamics problem of the ST robot. Several input trajectories are considered to investigate the different behavior for the center of the star. For each of the trajectories, three different groups of mode shapes are considered and their vibrational responses are compared. In this research, for the first time, effects of the passive prismatic joint parameters such as mass, rotational moment of inertia, and its actual length are considered in an analytical model. Finally, the analytical solution and a FEM (Finite Element Method) software solution are compared.