A theory, in the form of a coupled system of reduced parabolic wave equations (equations (42)), is developed for stress wave propagation studies through inhomogeneous, locally isotropic, linearly elastic solids. A parabolic wave theory differs from a complete wave theory in allowing propagation only in directions of increasing range. Thus, when applicable, it is well suited for numerical computation using a range-incrementing procedure. The parabolic theory considered here requires the propagation directions to be limited to a cone, centered about a principal propagation direction, which might be described as narrow-angled. Further, the theory requires that the effects of diffraction, refraction, and energy transfer between the dilatational and distortional modes are gradual enough that coupling between them can be ignored over a range of several wavelengths. Precise conditions for the applicability of the theory are summarized in a series of inequalities (equations (44)).