"My mother and my father went to eat soup."
Translation:Minha mãe e meu pai foram tomar sopa.
It depends on whether the previous verb demands a preposition (which may not be "a", it could be "de" or "por", etc). Here the previous verb is "ir" and you will recall from phrases like "Eu vou cantar" (I will sing) that it does not require a preposition when followed by a verb in the infinitive.
A good example of a verb that demands "a" is "começar" (to start) and you would have to write "Meus pais começaram a tomar sopa".