We introduce a concept for efficient optical parametric amplification (OPA) based on simultaneously phase-matched idler second harmonic generation (SHG), which together exhibit the dynamical behavior of parametric amplification but with damped conversion-back-conversion cycles. This enables amplification efficiency exceeding that of conventional OPA by several-fold for femtosecond and picosecond signal pulses with bell-shaped intensity profiles by allowing a near-uniform spatiotemporal depletion of the pump wave. We develop a Duffing oscillator model that unifies the description of conventional OPA and amplification accompanied by idler photon displacement by either linear absorption or SHG. A spatiotemporal analysis of devices based on birefringent or superlattice quasi-phase matching in common bulk media predicts energy conversion up to 55%.