Those are good points. Your initial comment criticized the endogeneity issue, not the multicollinearity. Further, controlling for parents income is essentially

Those are good points. Your initial comment criticized the endogeneity issue, not the multicollinearity. Further, controlling for parents income is essentially an AR1 model, which doesn't suffer from the multicollinearity interpretation issues you described. The problem would be if child willpower were too highly correlated with parents' income, not if parents' income were highly correlated with child's income.