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Oxidative stress is a condition caused by an imbalance in the occurrence of reactive oxygen species in the cells and tissues of organisms. An ultra-performance liquid chromatography–electrospray ionization tandem mass spectrometry (UPLC–ESI–MS/MS) method was developed for the simultaneous determination of two oxidative stress biomarkers, 8-hydroxydeoxyguanosine (8OHDG) and dityrosine (DIY), in the gills, skin, dorsal fin, and liver tissue of Atlantic salmon (Salmo salar) parr. The use of target analyte-specific 13C and 15N internal standards allowed quantification of each target analyte to be performed through the standard solvent calibration curve. The relative recoveries [mean ± (relative standard deviation%)] of 8OHDG and DIY were 101 ± 11 and 104 ± 13% at a fortified concentration of 10 ng/mL (8OHDG) and 500 ng/mL (DIY), respectively, ensuring the accuracy of the extraction and quantification. The chromatographic separation was carried out using a gradient elution program with a total run time of 5 min. The limits of detection (LODs) were 0.11 and 1.37 ng/g wet weight (w.w.) for 8OHDG and DIY, respectively. To demonstrate the applicability of the developed method, it was applied in 907 tissue samples that were collected from Atlantic salmon parr individuals reared in an experimental land-based recirculating aquaculture system (RAS) treated with peracetic acid. Moreover, the possibility of using the dorsal fin as an alternative matrix for the minimally invasive assessment of oxidative stress in Atlantic salmon parr was introduced. To our knowledge, 8OHDG and DIY were used for the first time as biomarkers for biomonitoring the fish health (oxidative stress) of Atlantic salmon parr in RAS.