My father often used to tell us not to envy others.
Literal
Father [topic-は] often us [to-に] others [object-を] don't-envy [quotative-と] was-saying.
うらやむな uses the negative imperative な: dictionary form + な = 'don't!' — the most direct prohibition. It's quoted with と inside reported speech. ~と言っていた describes a habitual past action of the father giving moral instructions.