My sister and I ___ while my dad ___.

The sentence describes two simultaneous actions in the past. The first action ('My sister and I ___') was ongoing when the second action ('my dad ___') interrupted or occurred during it. Therefore, the past continuous tense is appropriate for both clauses. The subject 'My sister and I' is plural, requiring 'were chatting'. The subject 'my dad' is singular, requiring 'was driving'. Thus, the correct option is 'were chatting / was driving'.