Recent microscopic studies have investigated various features of the spin distributions of fission fragments, but their parity distributions remain largely unexplored. In this Letter, we provide a complete characterization of the spin–parity content of fission fragments within a time-dependent Hartree–Fock–Bogoliubov framework, performing for the first time simultaneous projections on angular momentum, particle number, and parity. Calculations are carried out for the thermal neutron-induced fission of ${}^{239}$Pu using both the Gogny and Skyrme energy density functionals. We find that dynamical pair breaking during fission populates a significant fraction of unnatural-parity states and generates components with non-zero spin projections $K$. The parity content is found to depend on the number parity of the fragments, with odd-mass nuclei exhibiting pronounced parity staggering and odd-odd nuclei favoring negative parity. These results show that the parity distribution of fragments can depart significantly from the equiprobable partition commonly assumed in statistical de-excitation models, with potential implications for the modeling of fragment decay.