Multi-beam-pumped optical parametric amplification (OPA) may offer a solution for overcoming the limitations on the repetition rate and average power of ultraintense femtosecond lasers. However, multi-beam pumping scheme encounters challenges due to low conversion efficiency caused by parametric diffraction and inter-channel crosstalk, which are attributed to the presence of multi-idler-waves. This paper introduces and explores multi-beam-pumped quasi-parametric amplification (QPA) as an alternative approach to multi-beam-pumped OPA. We theoretically demonstrate that strong idler dissipation can significantly reduce inter-channel crosstalk and parametric diffraction in a two-beam-pumped QPA. These findings highlight the potential of multibeam-pumped QPA for efficiently amplifying femtosecond laser pulses to high average power levels with robustness.