Resolution of the intrinsic conflict between the reproduction of single cells and the homeostasis of a multicellular organism is central to animal biology and has direct impact on aging and cancer. Intercellular competition is indispensable in multicellular organisms because it weeds out senescent cells, thereby increasing the organism's fitness and delaying aging. In this Letter, we describe the growth dynamics of multicellular organisms in the presence of intercellular competition and show that the lifespan of organisms can be extended and the onset of cancer can be delayed if cells alternate between competition (a fair strategy) and noncompetitive growth, or cooperation (a losing strategy). This effect recapitulates the weak form of the game-theoretic Parrondo's paradox, whereby strategies that are individually fair or losing achieve a winning outcome when alternated. We show in a population model that periodic and stochastic switching between competitive and cooperative cellular strategies substantially extends the organism lifespan and reduces cancer incidence, which cannot be achieved simply by optimizing the competitive ability of the cells. These results indicate that cells could have evolved to optimally mix competitive and cooperative strategies, and that periodic intercellular competition could potentially be exploited and tuned to delay aging.