One inherent characteristic of quadratic nonlinear interaction is that it allows both forward and backward energy transfer among the three interacting waves. This backconversion effect, universal in all the parametric processes, is detrimental when a unidirectional energy transfer is desired and limits the conversion efficiency. We report a family of quadratic nonlinear interactions, quasi-parametric amplification (QPA), in which the idler wave is depleted by the introduction of a material loss and only the signal is amplified. In contrast to optical parametric amplification (OPA), the QPA scheme can inhibit the backconversion effect and thus enable ideal chirped-pulse amplification with high conversion efficiency and broad gain bandwidth. We have numerically proved the feasibility of this new scheme, and experimentally realized it by using a ${\mathrm{Sm}}^{3+}$-doped yttrium calcium oxyborate crystal that is highly absorptive at the idler wavelength and transparent at the pump and signal wavelengths. Amplification of broadband chirped pulses, corresponding to a pump depletion of 70% and a signal efficiency of 41%, has been achieved in a typical Gaussian pump case, exceeding the results of the previously reported state-of-the-art OPA. The proposed QPA scheme will be a promising approach for efficiently amplifying chirped pulses to unprecedented powers.