In this paper, the conditions of the self-similar propagation of the parabolic pulse (PP) in passive nonlinear tapers are derived and classified into the three cases. Analytical solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE) for the three cases are obtained. Numerical verifications of these analytical solutions are demonstrated by designing the silicon waveguide tapers. Moreover, we further design three kinds of cascaded silicon waveguides for realizing the PP generation and compression. The PP compression occurs at the anomalous-dispersion segment of the cascaded silicon waveguide. The generalized INLSE is used to model the generation and compression of the PP. When considering the higher order dispersion, higher order nonlinearity, and losses, the complex nonlinear dynamics are investigated. Simulation results show that a 300-fs Gaussian input pulse can evolve to the PP and be compressed to 35.6 fs in the cascaded silicon waveguide.