In this paper, instability due to occurrence of parametric resonance in transverse vibration of a rectangular plate on an elastic foundation under passage of continuous series of moving masses is examined as a model of bridge-moving loads interaction. The extended Hamilton’s principle is employed to derive the partial differential equation of motion. Subsequently, the governing partial differential equation is transformed into a set of ordinary differential equations by the Galerkin procedure. Considering local, Coriolis and centripetal acceleration components of the moving masses in the analysis leads to appearance of time-varying mass, damping and stiffness matrices in the coefficients of the governing equation. The passage of continuous series of moving masses along the rectilinear path results in a parametrically excited system with periodic coefficients. Applying incremental harmonic balance method as a semi-analytical method to the governing equations, stability of the system is investigated for a wide range of masses and velocities of the passing loads and different boundary conditions of the plate. Moreover, effect of the foundation stiffness on stability of the plate is examined. Results indicate that using clamped supports for the edges of entrance and departure of masses over the plate’s surface leads to formation of an instability tongue in the parameters plane which does not appear for the case of using simply supports. Also, it is observed that critical velocities of the moving masses will be increased by escalation the foundation stiffness. Numerical simulations confirm the accuracy of the semi-analytical results.