An interface & curvature reconstruction method on unstructured meshes is presented by generalizing the ideas and efforts expressed in a previous work (cf. Kromer et al. [4]) the author has contributed to. Herein, a novel piecewise linear interface (re-)construction method, henceforth referred to as PLIC, has been proposed, whose concept is extended to accommodate a piecewise parabolic interface (re-)construction method, abbreviated as PPIC. Analogously to PLIC, the overall strategy will be to fit a paraboloid to the volume fraction data in a least-squares sense. The resulting error sum is then read as a function that is parametrized by the paraboloids base point, orientation and curvature, which allows to apply the Reynolds transport theorem to retrieve derivative information for a Newton-type minimization scheme. As it was vital to PLIC, the optimization problem that amounts to PPIC will be complemented by suitable constraints to account for volume conservation and stability.