We investigate the behavior of persistent currents for a fixed number of noninteracting fermions in a periodic quantum ladder threaded by Aharonov-Bohm and transverse magnetic fluxes 􏰀 and χ . We show that the coupling between ladder legs provides a way to effectively change the ground-state fermion-number parity, by varying χ . Specifically, we demonstrate that varying χ by 2π (one flux quantum) leads to an apparent fermion-number parity switch. We find that persistent currents exhibit a robust 4π periodicity as a function of χ, despite the fact that χ → χ + 2π leads to modifications of order 1/N of the energy spectrum, where N is the number of sites in each ladder leg. We show that these parity-switch and 4π periodicity effects are robust with respect to temperature and disorder, and outline potential physical realizations using cold atomic gases and photonic lattices, for bosonic analogs of the effects.