<p>The non-parametric Budyko framework provides empirical relationships between a catchment's long-term mean evapotranspiration (<span style="border-top: 1px solid #000; color: #000;"><i>E</i></span>) and the aridity index, defined as the ratio of mean rainfall depth (<span style="border-top: 1px solid #000; color: #000;"><i>P</i></span>) to mean potential evapotranspiration (<span style="border-top: 1px solid #000; color: #000;"><i>E</i><sub>0</sub></span>). The parametric Budyko equations attempt to generalize this framework by introducing a catchment-specific parameter (<i>n</i> or <i>w</i>), intended to represent differences in catchment climate and landscape features. Many studies have developed complex regression relationships for the catchment-specific parameter in terms of biophysical features, all of which use known values of <span style="border-top: 1px solid #000; color: #000;"><i>P</i></span>, <span style="border-top: 1px solid #000; color: #000;"><i>E</i><sub>0</sub></span>, and <span style="border-top: 1px solid #000; color: #000;"><i>E</i></span> to numerically invert the parametric Budyko equations to obtain values of <i>n</i> or <i>w</i>. In this study, we analytically invert both forms of the parametric Budyko equations, producing expressions for <i>n</i> and <i>w</i> only in terms of <span style="border-top: 1px solid #000; color: #000;"><i>P</i></span>, <span style="border-top: 1px solid #000; color: #000;"><i>E</i><sub>0</sub></span>, and <span style="border-top: 1px solid #000; color: #000;"><i>E</i></span>. These expressions allow for <i>n</i> and <i>w</i> to be explicitly expressed in terms of biophysical features through the dependence of <span style="border-top: 1px solid #000; color: #000;"><i>P</i></span>, <span style="border-top: 1px solid #000; color: #000;"><i>E</i><sub>0</sub></span>, and <span style="border-top: 1px solid #000; color: #000;"><i>E</i></span> on those same features.</p>