Biomass and fossil fuels are mostly used in rural areas of developing countries for cooking. These energy resources have negative impacts on the environment and human health mainly due to deforestation and greenhouse gas emissions. A cleaner and environmentally-friendly form of cooking is solar cooking using the sun's energy. Parabolic solar cookers perform better than other solar cookers such as box solar cookers since higher temperatures are achieved in a shorter duration. These higher temperatures allow most types of cooking processes such as; boiling, frying, roasting, and baking to be possible. The major problem with most conventional solar cookers is that cooking is not possible during off-sunshine periods. Integrating solar cookers with thermal energy storage (TES) makes cooking during off-sunshine periods possible. This paper presents a comprehensive review of parabolic solar cookers with TES which is a sustainable cooking solution for developing countries. Previous work on both parabolic trough and parabolic dish solar cookers with TES is presented. Solar cookers with storage are classified according to the two main types of TES technologies which are; sensible heat thermal energy storage (SHTES) and latent heat thermal energy storage (LHTES). The main conclusions of the review are that; parabolic dish solar cookers with TES are more common than parabolic trough cookers, more studies have been done using latent heat storage as compared to sensible heat storage, limited modelling work has been presented, most of the parabolic cookers used aluminum sheets as the reflecting material, large scale experimental studies are limited and fewer studies tackled the techno-economic and socio-economic aspects. The review also highlights research gaps on parabolic solar cookers with TES in terms of the modelling, combined TES (sensible and latent heat), different reflector materials, large-scale experimental setups, and techno-economic and socio-economic analyses. Other important aspects to consider for parabolic solar cookers with TES include; the choice of sensible or latent heat storage materials, environmental impact, types of food to be cooked, and social acceptance.