Why are omega-3 fatty acids so easily oxidized when they're incorporated in cellular membranes?
I think the explanation for this description of fish oils as "easily oxidised" can be found in the Introduction to the actual paper (Nutrition 27 (2011) 334–337) that is cited in the article linked to in the question.


  Fish oil contains high levels of eicosapentaenoic acid (EPA; 20:5 omega-3) and docosahexaenoic acid (DHA; 22:6 omega-3), which are omega-3 polyunsaturated fatty acids. EPA, DHA, and fish oil have been shown to have protective effects against coronary heart disease, thrombosis, inflammatory processes, carcinomatosis, and metabolic syndrome. Therefore, fish oils and components of the oils are marketed as health supplements. However, the effects of omega-3 polyunsaturated fatty acids on aging and lifespan are unclear compared with those of omega-6 polyunsaturated fatty acids contained in safflower oil and soybean oil. EPA and DHA are oxidized easily compared with linoleic acid (18:2 omega-6) and oleic acid (18:1 omega-9) in vitro . 


In other words, the fish oil omega-3 fatty acids EPA and DHA have, respectively, 5 and 6 double bonds, whereas the omega-6 fatty acids in safflower oil, linoleic acid and oleic acid have, respectively, just 2 and 1 double bonds. So the fatty acids in fish oil are more prone to oxidation simply because they have so many more double bonds.

Incidentally, as you can see from the two structures below, both of the omega-3 fatty acids, EPA and DHA, also have an omega-6 double bond.

EPA:


DHA: