We propose an explanation for the electronic nematic state observed recently in parent iron-based superconductors [T.-M. Chuang, Science 327, 181 (2010)10.1126/science.1181083]. We argue that the quasi-one-dimensional nanostructure identified in the quasiparticle interference (QPI) is a consequence of the interplay of the magnetic (π, 0) spin-density wave (SDW) order with the underlying electronic structure. We show that the evolution of the QPI peaks largely reflects quasiparticle scattering between bands involved in the SDW formation. Because of the ellipticity of the electron pocket and the fact that only one of the electron pockets is involved in the SDW, the resulting QPI has a pronounced one-dimensional structure. We further predict that the QPI crosses over to two dimensionality on an energy scale, set by the SDW gap.