The self-similarity techniques are used to study pulse propagation in a quantum dot waveguide amplifier with an arbitrary longitudinal gain profile. Generation of high-power parabolic pulses are analyzed under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. In particular, the periodic distributed gain enables us to obtain a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp. Finally, a possible experimental protocol is proposed to generate the high-power parabolic pulses in realistic waveguides.