Birds and mammals share specialized forms of sleep including slow wave sleep (SWS) and rapid eye movement sleep (REM), raising the question of why and how specialized sleep evolved. Extensive prior studies concluded that avian sleep lacked many features characteristic of mammalian sleep, and therefore that specialized sleep must have evolved independently in birds and mammals. This has been challenged by evidence of more complex sleep in multiple songbird species. To extend this analysis beyond songbirds, we examined a species of parrot, the sister taxon to songbirds. We implanted adult budgerigars (Melopsittacus undulatus) with electroencephalogram (EEG) and electrooculogram (EOG) electrodes to evaluate sleep architecture, and video monitored birds during sleep. Sleep was scored with manual and automated techniques, including automated detection of slow waves and eye movements. This can help define a new standard for how to score sleep in birds. Budgerigars exhibited consolidated sleep, a pattern also observed in songbirds, and many mammalian species, including humans. We found that REM constituted 26.5% of total sleep, comparable to humans and an order of magnitude greater than previously reported. Although we observed no spindles, we found a clear state of intermediate sleep (IS) similar to non-REM (NREM) stage 2. Across the night, SWS decreased and REM increased, as observed in mammals and songbirds. Slow wave activity (SWA) fluctuated with a 29-min ultradian rhythm, indicating a tendency to move systematically through sleep states as observed in other species with consolidated sleep. These results are at variance with numerous older sleep studies, including for budgerigars. Here, we demonstrated that lighting conditions used in the prior budgerigar study-and commonly used in older bird studies-dramatically disrupted budgerigar sleep structure, explaining the prior results. Thus, it is likely that more complex sleep has been overlooked in a broad range of bird species. The similarities in sleep architecture observed in mammals, songbirds, and now budgerigars, alongside recent work in reptiles and basal birds, provide support for the hypothesis that a common amniote ancestor possessed the precursors that gave rise to REM and SWS at one or more loci in the parallel evolution of sleep in higher vertebrates. We discuss this hypothesis in terms of the common plan of forebrain organization shared by reptiles, birds, and mammals.