Based on the classical laminated plate theory and von Kármán-type equations, the nonlinear governing equations of motion for the composite laminated plates under axial loading, in which, the boundary damping is taken into consideration, are derived. Galerkin approach is employed to transform the governing partial differential equations to ordinary differential equations. Using the method of multiple scales, the approximate solution of the plates by deterministic parametric resonance excitation are obtained and analyzed. Results show that the unstable interval bandwidth for the trivial solution is only related with linear damping, whereas the boundary nonlinear damping can weaken the resonance amplitude of the non-trivial solution and reduce the corresponding resonance range effectively. On the basis of nonlinear dynamic model, the resulted Foker-Planck-Kolmogorov equation is analyzed by using the narrow-band random excitation combined with finite difference method. The stochastic bifurcation and jump phenomenon of the response for the amplitude stationary joint probability density caused by the boundary nonlinear damping are sequentially investigated. Results show that a very small increment of the damping will induce a sharp jumping of the stationary motion from the non-trivial solution to the trivial one.