Generalized local and multi-dimensional conservation laws of action, energy, momentum, and angular momentum are derived for stimulated Raman (SRS) and Brillouin backscattering (SBS) in a density gradient within the paraxial ray approximation. A Lagrangian density is given for which requiring the action be stationary reproduces the well known envelope equations for SRS and SBS in density gradients in the absence of damping. Using Noether’s theorem, the symmetries of the Lagrangian density are used to obtain local conservation laws for quantities that can easily be identified as the wave action, energy, and momentum. These multi-dimensional conservation laws reduce to the well known one dimensional Manley-Rowe relations, and frequency and wavenumber matching conditions. Additional symmetries of the action lead to conservation laws for new quantities that are identified as the orbital angular momentum and additional contributions to the energy and momentum of the wave from frequency and wavenumber shifts. Augmentation of the conservation laws in the presence of damping and extensions of the Lagrangian to include higher order corrections to the paraxial ray equation and nonlinear frequency shifts are also provided.