A Laplace-domain boundary element approach for transient dynamic analysis of three-dimensional (3D) moderately thick multilayered (piecewise homogeneous) anisotropic linear elastic composite plates is presented. The boundary element formulation is based on the system of weakly singular displacement boundary integral equations. The spatial discretization is based on collocation method and mixed representation of geometry and boundary functions. To obtain time-domain solutions, the Convolution Quadrature Method with the Runge-Kutta method as an underlying time stepping method is used as a numerical technique for inverse Laplace transform. To improve the computational efficiency of the boundary element formulation a parallelization scheme is implemented. Boundary element results for the test example are provided to validate the proposed approach.