Currently, the utilization of hyperprolific sows has stimulated the search for supplements aimed at enhancing piglet survival, as these sows yield more offspring than they can adequately feed with their colostrum production. In contrast, intensive goat farming often yields surplus colostrum, thus necessitating its removal, since kids are exclusively fed colostrum through lactation solely within the initial day of birth. The objective of this study was to examine and compare the fatty acid (FA) profiles of colostrum from sows and goats, together with possible influencing factors such as sow parity and the postpartum day of the goat, for possible use as an energy supplement for neonatal piglets. Swine colostrum was collected from sows with a 0–5 parity. In addition, samples of goat colostrum were collected on their first (D1) and second (D2) days of postpartum milking. The FA profiles of the colostrum were analyzed via gas chromatography. The parity value of the sows did not affect (p > 0.05) the FA colostrum composition. High proportions of palmitic, oleic, and linoleic acids were found in both types of colostrum. Levels of palmitic, oleic, and linoleic acids were significantly higher in D1 goat colostrum, whereas saturated FAs of less than 14 carbons (4:0, 6:0, 8:0, 10:0, and 12:0) were found in higher proportions in D2. These FAs play an important role in colostrum as they are a readily available source of energy and have also been attributed strong antibacterial activity. Therefore, goat colostrum, especially D2, could be used as an alternative energy supplement for newborn piglets, in particular for the weakest and smallest of the litter, which are the most in need.