We infer the role of gender identity norms from the reallocation of childcare across parents, following changes in their relative wages. By exploiting variation from a Swedish tax reform, we estimate the elasticity of substitution in parental childcare for the whole population and for demographic groups potentially adhering to differently binding norms. We find that immigrant, married and male breadwinner couples, as well as couples with a male first-born, react more strongly to tax changes that induce a more traditional allocation of spouses' time, while the respective counterpart couples react more strongly to tax changes that induce a more egalitarian division of labor.