The quantum Hall states at filling factorsν=5/2and7/2are expected to have Abelian charge-e/2quasiparticles and non-Abelian charge-e/4quasiparticles. The non-Abelian statistics of the latter is predicted to display a striking interferometric signature, the even-odd effect. By measuring resistance oscillations as a function of the magnetic field in Fabry-Pérot interferometers using new high-purity heterostructures, we for the first time report experimental evidence for the non-Abelian nature of excitations atν=7/2. At bothν=5/2and7/2, we also examine, for the first time, the fermion parity, a topological quantum number of an even number of non-Abelian quasiparticles. The phase of observede/4oscillations is reproducible and stable over long times (hours) near both filling factors, indicating stability of the fermion parity. At both fractions, when phase fluctuations are observed, they are predominantlyπphase flips, consistent with either fermion parity change or change in the number of the enclosede/4quasiparticles. We also examine lower-frequency oscillations attributable to Abelian interference processes in both states. Taken together, these results constitute new evidence for the non-Abelian nature ofe/4quasiparticles; the observed lifetime of their combined fermion parity further strengthens the case for their utility for topological quantum computation.