When my child causes trouble for another child, how far should the parent apologize?

There's no clear line—it depends on the situation. In common: if there's injury or discomfort to the other side, sincerely apologizing and showing care first matters. But there's no need to keep apologizing endlessly to the point of wearing yourself out. As much as apology, sharing "why it happened and how to prevent it" with the preschool/school and reducing recurrence is constructive. What the family decides: how far to apologize or explain is fine to choose by the relationship and situation. A guide: "did I convey care for safety and an apology?" "did I share prevention with those involved?" Apologizing isn't the goal—creating a form where everyone can spend time securely is.