Periodic trains of mesoscale structures in solar wind density have been observed close to the Sun with in situ data from the Helios spacecraft, as well as remotely in STEREO/COR2 and STEREO/HI1 white light imaging data. While some periodic density structures may be a consequence of the development of dynamics en route, many are remnants of the formation and release of the solar wind, and thus provide important constraints on solar wind models. The instrument suite on Parker Solar Probe offers an unprecedented viewpoint of the ambient solar wind and structure therein, shortly after its formation and release from the solar corona. Here, we report on the first observations of periodic trains of mesoscale structures in solar wind density observed by the Wide-field Imager for Parker Solar PRobe (WISPR). We describe our open-source Fourier analysis and robust spectral background estimation technique used to identify the periodic density structures. The observation of periodic density structures so near to the Sun allows us to begin disentangling how much structure is created during solar wind formation, versus how much is due to evolution as the solar wind advects outward.