A theory of parametric interaction between spin waves localized in a waveguide and traveling elastic waves is developed for ferromagnetic thin films. The presented theoretical formalism takes into account an arbitrary spatial distribution of the displacement field in the acoustic waves and an arbitrary magnetization in spin waves. Using the theory, we examine interaction of forward-volume spin waves (FVSW) localized in a narrow waveguide and Rayleigh surface acoustic waves traveling in a substrate underneath the waveguide. We show that, in contrast to classical electromagnetic pumping, the symmetry of the magnetoelastic interaction allows for the generation of first-order parametric instabilities in spin waves with circular precession, such as FVSW. At the same time the localization of spin waves modifies the momentum conservation law for the parametric process to include the transfer of momentum to the waveguide, which allows for a frequency separation of the interacting counterpropagating spin waves. The frequency separation enables amplification of a localized spin wave without generation of a traveling idler wave, which results in a greater amplification efficiency.