Open clusters (OCs) provide homogeneous samples of white dwarfs (WDs) with known distances, extinctions, and total ages. The unprecedented astrometric precision of Gaia allows us to identify many novel OC-WD pairs. Studying WDs in the context of their parent OCs makes it possible to determine the properties of WD progenitors and study the initial-final mass relation (IFMR). We seek to find potential new WD members of OCs in the solar vicinity. The analysis of OC members' parallaxes allows us to determine the OC distances to a high precision, which in turn enables us to calculate WD masses and cooling ages and to constrain the IFMR. We searched for new potential WD members of nearby OCs using the density-based machine learning clustering algorithm HDBSCAN. The clustering analysis was applied in five astrometric dimensions-positions in the sky, proper motions and parallaxes, and in three dimensions where the positional information was not considered in the clustering analysis. The identified candidate OC WDs were further filtered using the photometric criteria and properties of their putative host OCs. The masses and cooling ages of the WDs were calculated via a photometric method using all available Gaia, Pan-STARRS, SDSS, and GALEX photometry. The WD progenitor masses were determined using the ages and metallicities of their host OCs. Altogether, 63 OC WD candidates were recovered, 27 of which are already known in the literature. We provide characterization for 36 novel WDs that have significant OC membership probabilities. Six of them fall into relatively unconstrained sections of the IFMR where the relation seems to exhibit nonlinear behavior. We were not able to identify any WDs originating from massive progenitors that would even remotely approach the widely adopted WD progenitor mass limit of 8M_{sun}_; this confirms the paucity of such objects residing in OCs and hints at a presence of velocity kicks for nascent WDs.