The Advanced LIGO observatories detected gravitational waves from two binary black hole mergers during their first observation run (O1). We present a high-energy neutrino follow-up search for the second gravitational wave event, GW151226, as well as for gravitational wave candidate LVT151012. We find two and four neutrino candidates detected by IceCube, and one and zero detected by Antares, within ±500 s around the respective gravitational wave signals, consistent with the expected background rate. None of these neutrino candidates are found to be directionally coincident with GW151226 or LVT151012. We use nondetection to constrain isotropic-equivalent high-energy neutrino emission from GW151226, adopting the GW event's 3D localization, to less than 2×1051-2×1054 erg.