Presently, there is an increasing shift towards the utilization of natural antioxidants and compounds with protective attributes for fatty acids in order to replace synthetic counterparts that may pose health risks. This transition aligns with the growing emphasis on promoting healthy and organic food choices. Essential oils stand out in this context due to scientific validations of their antioxidant properties. There are few published research results concerning changes in the fatty acid composition in model systems with the addition of essential oils. This study aims to investigate the impact of incorporating peppermint and thyme oils on inhibiting changes in the fatty acid profile of sunflower oil stored at both room temperature with exposure to daylight and in a thermostat set at 40 °C. The experimental procedure involved the addition of peppermint and thyme oils, along with butylated hydroxyanisole (BHA), to batches of sunflower oil. The samples were then stored for 11 months. The study observed a detrimental influence of storage conditions on the quantitative changes in the fatty acid profile of the sunflower oil. The addition of BHA stabilized the content of linoleic acid in the sunflower oil (approximately 53 g/100 g of linoleic acid compared to approximately 58 g/100 g in the control sample). Meanwhile, the model system of sunflower oil with the addition of peppermint and thyme oils (40 °C) exhibited a statistically significant decrease in the concentration of linoleic acid to approximately 8 g/100 g after eleven months of thermostating. Similar trends to those observed for linoleic acid were noted for the total fatty acid content in the sunflower oil. Notably, the efficacy of the selected substances in inhibiting adverse transformations in fats was contingent upon their concentration and the storage temperature.