Quasi-one-dimensional magnets can host an ordered longitudinal spin-density wave state (LSDW) in magnetic field at low temperature, when longitudinal correlations are strengthened by Ising anisotropies. In the S=1/2 Heisenberg antiferromagnet YbAlO_{3} this happens via Ising-like interchain interactions. Here, we report the first experimental observation of magnetization plateaux at 1/5 and 1/3 of the saturation value via thermal transport and magnetostriction measurements in YbAlO_{3}. We present a phenomenological theory of the plateau states that describes them as islands of commensurability within an otherwise incommensurate LSDW phase and explains their relative positions within the LSDW phase and their relative extent in a magnetic field. Notably, the plateaux are stabilized by ferromagnetic interchain interactions in YbAlO_{3} and consistently are absent in other quasi-1D magnets such as BaCo_{2}V_{2}O_{8} with antiferromagnetic interchain interactions. We also report a small, steplike increase of the magnetostriction coefficient, indicating a weak phase transition of unknown origin in the high-field phase just below the saturation.