Fruit kernels are bio-waste in the food industry and can be managed as oil raw materials. The aim of the study was to investigate the stability, oxidation resistance, profile of fatty acids, and their distribution at the triacylglycerol positions of commercially available cold-pressed oils from apricot (A1 and A2) and peach kernels (P1 and P2). Additionally, the melting profiles of the oils were recorded. It was found that the tested oils were a good source of unsaturated fatty acids, especially oleic and linoleic acid. The apricot kernel oils differed significantly regarding the contents of oleic acid and linoleic acid, while the peach kernel oils did not differ in the content of fatty acids. Saturated palmitic and stearic acids mainly occupied the external sn-1,3 positions, while linoleic acid was esterified in the internal position in all the analyzed oils. The occurrence of endothermic peaks at low temperatures was recorded on the melting curves, indicating the presence of triacylglycerol fractions with high proportions of poly- and monounsaturated fatty acids. The apricot kernels of the A1 oil were characterized by the highest oxidative stability measured by the maximum oxidation time at the tested temperatures. The activation energy of the oxidation reaction was at a similar level in all the studied oils. Due to the oils’ characteristics, it is worth considering the use of apricot and peach kernel oils in the food industry.