The Gaia Data Release 1 (DR1) sample of white dwarf parallaxes is presented, including six directly observed degenerates and 46 white dwarfs in wide binaries. This data set is combined with spectroscopic atmospheric parameters to study the white dwarf mass-radius relationship (MRR). Gaia parallaxes and G magnitudes are used to derive model atmosphere-dependent white dwarf radii, which can then be compared to the predictions of a theoretical MRR. We find a good agreement between Gaia DR1 parallaxes, published effective temperatures (T_eff_) and surface gravities (logg), and theoretical MRRs. As it was the case for Hipparcos, the precision of the data does not allow for the characterization of hydrogen envelope masses. The uncertainties on the spectroscopic atmospheric parameters are found to dominate the error budget and current error estimates for well-known and bright white dwarfs may be slightly optimistic. With the much larger Gaia DR2 white dwarf sample, it will be possible to explore the MRR over a much wider range of mass, T_eff_, and spectral types.