We demonstrate enhanced conversion efficiency (CE) and parametric amplification of optical pulses via quasiphase- matched four-wave-mixing (FWM) in long-period Bragg waveguides made of silicon. Our study is based on a rigorous theoretical model that describes optical pulse dynamics in a periodically, adiabatically modulated silicon photonic waveguide and a comprehensive set of numerical simulations of pulse interaction in such gratings. More specifically, our theoretical model takes into account all of the relevant linear and nonlinear optical effects, including free-carriers generation, two-photon absorption, and self-phase modulation, as well as modal frequency dispersion up to the fourth-order. Due to its relevance to practical applications, a key issue investigated in our work is the dependence of the efficiency of the FWM process on the waveguide parameters and the operating wavelength. In particular, our analysis suggests that by varying the waveguide width by just a few tens of nanometers the wavelengths of the phase-matched waves can be shifted by hundreds of nanometers. Our numerical simulations show also that, in the anomalous group-velocity dispersion regime, a CE enhancement of more than 20 dB, as compared to the case of a waveguide with constant width, can be easily achieved.