Learn Italian: Parte senza che suo padre gli dia il permesso. - He leaves without his father giving him permission.

Dia is the present subjunctive form of dare (to give). We use the subjunctive here because the phrase is introduced by the conjunction senza che (without). This conjunction always triggers the subjunctive because it describes an action that is not happening.

Could I just say senza dare instead of senza che... dia?

No, because the subjects of the two actions are different. The person leaving is he (parte), but the person giving permission is his father (suo padre). When the subjects are different, you must use senza che followed by a conjugated subjunctive verb. You only use senza plus an infinitive when the subject is the exact same for both actions.

Why is there no article before suo padre, like il suo padre?

In Italian, you do not use definite articles like il or la with possessive adjectives when referring to a singular, unmodified family member. Since padre (father) is a singular family member, it is simply suo padre.

What role does gli play in this sentence?

Gli is the indirect object pronoun meaning 'to him'. It shows who is receiving the permission. Even though the verb dia is in the subjunctive mood, the rule for pronoun placement stays the same: it goes directly before the conjugated verb.