We introduce the notion of dissipative optical parametric amplifiers (DOPA) and demonstrate that, even in the absence of the Hermitian phase-matching condition in these structures, the signal beam can be amplified when the idler mode suffers optical attenuation. We discuss the optical implementation of this concept in waveguide platforms, and we propose different methods to control the optical loss of these configurations only at the wavelength of the idler component. Surprisingly, this spectrally selective dissipation process allows the signal beam to draw more energy from the pump and, as a result, attains net amplification. Similar results also apply if the losses are introduced only to the signal component. This intriguing feature can open new avenues for building long wavelength light sources and parametric amplifiers by using semiconductor planar structures, where Hermitian phase-matching requirements can be difficult to satisfy without adding stringent geometric constraints or relatively complex fabrication steps.