Which geometrical or material features determine the quality of an instrument is probably the most debated matter among guitar makers, but also one that is seldom approached with a scientific mindset. Numerical simulations have proven to be an effective tool for providing rigorous answers to such questions, as they facilitate the testing of multiple models within a shorter time frame, while still providing highly accurate results. However, these approaches are limited by the availability of instrument models which can be adjusted on the fly without excessive effort, possibly even in an automated way. In this perspective, the advantage offered by the use of parametric models, which has recently received particular attention in research, becomes evident. In this work we develop a parametric and CAD native model of a complete guitar, enabling convenient control over the geometrical features which are considered key in the context of guitar making. In particular, we focus on the parameterization of the outline shape and the soundboard bracing. Additionally, we utilize our new parametric model to conduct a set of simulations based on the Finite Element Method. By doing so, we investigate the effect of some common geometrical variations when these are applied on a full guitar model, including the effect of the surrounding air. The effect is evaluated both by looking at the eigenfrequency variations and at the frequency response of the instrument, in the form of its bridge mobility and radiated sound. Our findings are both in line with previous studies, proving the validity of our approach, but also present some novel results. In particular, we demonstrate that the uppermost transverse bars of the soundboard do not play any significant role in either shaping the instrument timbre or improving its radiation efficiency. Most importantly, we show how our model can be used to facilitate future research on the physics of guitars.