Collective inertia is strongly influenced at the level crossing at which the quantum system changes its microscopic configuration diabatically. Pairing correlations tend to make the large-amplitude nuclear collective motion more adiabatic by reducing the effect of these configuration changes. Competition between pairing and level crossing is thus expected to have a profound impact on spontaneous fission lifetimes. To elucidate the role of nucleonic pairing on spontaneous fission, we study the dynamic fission trajectories of 264Fm and 240Pu using the state-of-the-art self-consistent framework. We employ the superfluid nuclear density functional theory with the Skyrme energy density functional SkM* and a density-dependent pairing interaction. Along with shape variables, proton and neutron pairing correlations are taken as collective coordinates. The collective inertia tensor is calculated within the nonperturbative cranking approximation. The fission paths are obtained by using the least action principle in a four-dimensional collective space of shape and pairing coordinates. Pairing correlations are enhanced along the minimum-action fission path. For the symmetric fission of 264Fm, where the effect of triaxiality on the fission barrier is large, the geometry of the fission pathway in the space of the shape degrees of freedom is weakly impacted by pairing. This is not the case for 240Pu, where pairing fluctuations restore the axial symmetry of the dynamic fission trajectory. The minimum-action fission path is strongly impacted by nucleonic pairing. In some cases, the dynamical coupling between shape and pairing degrees of freedom can lead to a dramatic departure from the static picture. As a result, in the dynamical description of nuclear fission, particle-particle correlations should be considered on the same footing as those associated with shape degrees of freedom.