We employ a set of Magneticum cosmological hydrodynamic simulations that span over \(15\) different cosmologies, and extract masses and concentrations of all well-resolved haloes between \(z=0-1\) for critical over-densities \(\Delta_\texttt{vir}, \Delta_{200c}, \Delta_{500c}, \Delta_{2500c}\) and mean overdensity \(\Delta_{200m}.\) We provide the first mass-concentration (Mc) relation and sparsity relation (i.e. \(M_{\Delta1} - M_{\Delta2}\) mass conversion) of hydrodynamic simulations that is modelled by mass, redshift and cosmological parameters \(\Omega_m, \Omega_b, \sigma_8, h_0\) as a tool for observational studies. We also quantify the impact that the Mc relation scatter and the assumption of NFW density profiles have on the uncertainty of the sparsity relation. We find that converting masses with the aid of a Mc relation carries an additional fractional scatter (\(\approx 4\%\)) originated from deviations from the assumed NFW density profile. For this reason we provide a direct mass-mass conversion relation fit that depends on redshift and cosmological parameters. We release the package hydro\_mc, a python tool that perform all kind of conversions presented in this paper.