Quantitative estimation of rock physics properties is an important part of reservoir characterization. Most current seismic workflows in this field are based on amplitude variation with offset. Building on recent work on high resolution multi-parameter inversion for reservoir characterization, we construct a rock-physics parameterized elastic full-waveform inversion (EFWI) scheme. Within a suitably-formed multi-parameter EFWI, in this case a 2D frequency-domain isotropic-elastic FWI with a truncated Gauss-Newton optimization, any rock physics model with a well-defined mapping between its parameters and seismic velocity/density can be examined. We select a three-parameter porosity, clay content, and water saturation (PCS) parameterization, and link them to elastic properties using three representative rock physics models: the Han empirical model, the Voigt-Reuss-Hill boundary model, and the Kuster and Toksöz inclusion model. Numerical examples suggest that conditioning issues, which make a sequential inversion (in which velocities and density are first determined through EFWI, followed by PCS parameters) unstable, are avoided in this direct approach. Significant variability in inversion fidelity is visible from one rock physics model to another. However, the response of the inversion to the range of possible numerical optimization and frequency selections, as well as acquisition geometries, varies widely. Water saturation tends to be the most difficult property to recover in all situations examined. This can be explained with radiation pattern analysis, where very low relative scattering amplitudes from saturation perturbations are observed. An investigation performed with a Bayesian approach illustrates that the introduction of prior information may increase the inversion sensitivity to water saturation