Bayesian inference has recently been demonstrated to be effective in estimating stationary and non-stationary reduced-order vocal fold model parameters, along with the associated levels of uncertainty, from simulated glottal area waveform measures (Hadwin et al., 2016, 2017). In these studies, the fitting model was a three mass body-cover model with two degrees of freedom in the cover layer. While demonstrative, restricting the fitting model to two degrees of freedom assumes a priori that this is sufficient to capture salient vocal fold dynamics, thus limiting future clinical applicability. To overcome this, we employ Bayesian inference to directly estimate tissue properties of a two-dimensional (2D) finite element vocal fold model from glottal area waveforms generated by both numerical simulations and recorded videos of synthetic vocal fold oscillations. We demonstrate that the 2D finite element model is not only capable of producing meaningful estimates with reasonable uncertainties, but is also capable of distinguishing between different experimental tissue properties, which is an essential step towards the development of patient specific models.