Nonlinear pulse dynamics in two stages of different active or passive fibers are investigated in this work. Numerical approach of the symmetrized split step Fourier method is used to solve the nonlinear Schrödinger equation in the presence of fiber gain, nonlinearity, and dispersion. An input Gaussian pulse evolves into a linearly chirped perfect parabolic pulse (PP) when it propagates through a standard normal dispersion decreasing fiber amplifier. At the same time, for an erbium-doped dispersion decreasing fiber amplifier with a similar dispersion variation with length, the semiparabolic pulse (SPP) is produced at the output end of the fiber. To our knowledge, this is shown for the first time. In second stage, the so-obtained perfect PP, SPP, and also a chirp-free perfect PP are fed into the input of several normal dispersion fibers and the comparative pulse evolution is studied in detail with the variations of dispersion coefficient, gain, and nonlinearity. While using these pulses as the input of an anomalous dispersion fiber, our result shows that the linearly chirped PP is most efficient for compressing the pulses with a good quality factor without dropping significant pedestal energy.