How did Lily's father get her to stop being angry at the boy at the playground?

Aside from being clueless about the world they live in Jonas's parents knew the fundamentals of social interactions. They ask Lily to consider that the boy was from a different district and may have had slightly different rules than Lily was used to.

The boy at the playground may have different rules because he lives in a different community. As does Lilly, and the children in her community.