Parabolic pulses have important applications in both basic and applied sciences, such as high power optical amplification, optical communications, all-optical signal processing, etc. The generation of parabolic similaritons in tapered hydrogenated amorphous silicon photonic wires at telecom (λ ~ 1550 nm) and mid-IR (λ ≥ 2100 nm) wavelengths is demonstrated and analyzed. The self-similar theory of parabolic pulse generation in passive waveguides with increasing nonlinearity is presented. A generalized nonlinear Schrödinger equation is used to describe the coupled dynamics of optical field in the tapered hydrogenated amorphous silicon photonic wires with either decreasing dispersion or increasing nonlinearity. The impacts of length dependent higher-order effects, linear and nonlinear losses including two-photon absorption, and photon-generated free carriers, on the pulse evolutions are characterized. Numerical simulations show that initial Gaussian pulses will evolve into the parabolic pulses in the waveguide taper designed.