In many animal species, parents provide care for their offspring, but the parental roles of the two sexes differ considerably between and within species. Here, we use an individual-based simulation approach to investigate the evolutionary emergence and stability of parental roles. Our conclusions are in striking contrast to the results of analytical models. In the absence of initial differences between the sexes, our simulations do not predict the evolution of egalitarian care, but either female-biased or male-biased care. When the sexes differ in their pre-mating investment, the sex with the highest investment tends to evolve a higher level of parental care; this outcome does not depend on non-random mating or uncertainty of paternity. If parental investment evolves jointly with sexual selection strategies, evolution results in either the combination of female-biased care and female choosiness or in male-biased care and the absence of female preferences. The simulations suggest that the parental care pattern drives sexual selection, and not vice versa. Finally, our model reveals that a population can rapidly switch from one type of equilibrium to another one, suggesting that parental sex roles are evolutionarily labile. By combining simulation results with fitness calculations, we argue that all these results are caused by the emergence of individual variation in parental care strategies, a factor that was hitherto largely neglected in sex-role evolution theory.