The grandparent comment explains how it still doesn't tell whether the parental involvement matters or not.

The experiment results show only that on average the kids perform better in better classrooms. But it's still possible that kids with better parental involvement have no benefit of it, and the statistical difference happens due to better results of just the kids with worse involvement.

In other words. Yes, the article controls for the fact that better parents can send kids to better kindergartens. It doesn't control for the fact that kids of better parents benefit from the better kindergartens.

It's possible that adding an interaction term for classroom quality and involved parent would show that there's zero marginal benefit to a good classroom if you're an involved parent. That result seems orthogonal to their research goal, which was estimating the mean impact of a quality classroom for the "average" child.

The phrasing, "they should control for" has a specific statistical meaning, criticizing the model as having an omitted variable problem -- that they left out a regressor that is correlated with both the dependant variable and at least one of the other regressors. That is a different criticism than suggesting an interaction term.