Over the past decades, hemodynamics simulators have become tools of choice for studying cardiovascular systems and are routinely used to simulate whole-body hemodynamics from physiological parameters. Nevertheless, solving the corresponding inverse problem of mapping waveforms back to plausible physiological parameters remains challenging.Motivated by advances in simulation-based inference (SBI), we cast this inverse problem as statistical inference. Our study highlights the potential of estimating new biomarkers from standard-of-care measurements and reveals practically relevant findings that cannot be captured by standard sensitivity analyses, such as the existence of sub-populations for which parameter estimation exhibits distinct uncertainty regimes. In addition, we study how such insights obtained in-silico transfer to in-vivo with the MIMIC-III database.