We report the molecular beam epitaxy growth of lateral composition-modulated (LCM) GaInP structures induced by GaP/InP short-period superlattice (SPS), and the effect of post-annealing process on their optical properties. The samples are structurally analyzed using cross-sectional transmission electron microscopy which reveals the formation of SPS-induced LCM-GaInP just above the substrate and bulk-like weakly phase separated LCM-GaInP towards the sample surface. The bulk-like features are formed due to strain accumulation and relaxation during SPS growth. The grown samples are annealed under different annealing conditions and optically characterized by low-temperature photoluminescence (LT-PL) and Raman spectroscopy. LT-PL reveals a lower energy PL peak originating from LCM-GaInP and a higher energy PL peak originating from weakly phase separated LCM-GaInP. These peaks red-shift with increase in annealing duration and temperature, and the higher energy PL peak approach towards the lower energy PL peak indicating enhanced phase separation of In-rich GaInP and Ga-rich GaInP vertical layers of the constituent LCM-GaInP structure. Also there exists an optimum annealing temperature and duration at which the PL intensity of the LCM-GaInP increases. More importantly, our results suggest that post-annealing process is a viable technique to enhance the optical properties of LCM-GaInP structures.