Much of the recent debate has centred on an ongoing controversy over whether primates are capable of reasoning about basic aspects of the visual perspective and perceptual awareness of others. One issue that has played a prominent role in this controversy concerns the relative "simplicity" or "parsimony" of mindreading and non-mindreading explanations of behavior. This chapter discusses the role that such appeal to simplicity have played in this controversy as a case study for thinking about the proper place of simplicity considerations in inferring cognitive processes in non-human animals, and in science more generally. After providing some background to the controversy and describing the appeals to simplicity that have been made, it poses some problems for such appeals. The chapter outlines a general philosophical account of simplicity. It sheds light on the nature of evidence for primate mindreading, and on how future work might resolve this controversy.