Hypertension (HTN) is the leading cause of premature deaths worldwide and the main preventable risk factor for cardiovascular diseases. Therefore, there is a current need for new therapeutics to manage this condition. In this regard, protein hydrolysates containing antihypertensive bioactive peptides are of increasing interest. Thus, agri-food industry byproducts have emerged as a valuable source to obtain these hydrolysates as they are rich in proteins and inexpensive. Among these, byproducts from animal origin stand out as they are abundantly generated worldwide. Hence, this review is focused on evaluating the potential role of chicken slaughterhouse byproducts as a source of peptides for managing HTN. Several of these byproducts such as blood, bones, skins, and especially, chicken feet have been used to obtain protein hydrolysates with angiotensin-converting enzyme (ACE)-inhibitory activity and blood pressure-lowering effects. An increase in levels of endogenous antioxidant compounds, a reduction in ACE activity, and an improvement of HTN-associated endothelial dysfunction were the mechanisms underlying their effects. However, most of these studies were carried out in animal models, and further clinical studies are needed in order to confirm these antihypertensive properties. This would increase the value of these byproducts, contributing to the circular economy model of slaughterhouses.