Superconductivity in the iron pnictides is often closely connected to a nematic state in which the tetragonal symmetry of the crystal is spontaneously broken. Determining the dominant interactions responsible for this symmetry breaking is essential to understanding the superconducting state. Here, we use atomic-resolution variable-temperature scanning tunnelling spectroscopy to probe the nanoscale electronic structure of the nematically ordered, parent pnictide NaFeAs and compare it with non-nematic LiFeAs. Local electronic nematicity is only manifest in NaFeAs and is found to persist to high temperatures in the nominally tetragonal phase of the crystal. The spatial distribution and energy dependence of the electronic anisotropy at high temperatures is explained by the persistence of large-amplitude, short-range, unidirectional, antiferroic fluctuations, indicating that strong density-wave fluctuations exist and couple to near-Fermi surface electrons even far from the structural and density-wave phase boundaries.