We study dephasing in electron transport through a large quantum dot (a Fabry-Perot interferometer) in the fractional quantum Hall regime with filling factor 2/3. In the regime of sequential tunneling, dephasing occurs due to electron fractionalization into counterpropagating charge and neutral edge modes on the dot. In particular, when the charge mode moves much faster than the neutral mode, and at temperatures higher than the level spacing of the dot, electron fractionalization combined with the fractional statistics of the charge mode leads to the dephasing selectively suppressing h/e Aharonov-Bohm oscillations but not h/(2e) oscillations, resulting in oscillation-period halving.