Learn French: Les parents pleurèrent quand leur fille épousa le roi. - The parents cried when their daughter married the king.

This is the passé simple (literary past tense) of the regular -er verb pleurer (to cry). For the third-person plural (they), regular -er verbs drop the -er and add -èrent. The grave accent is simply part of this specific ending. You will mostly see this tense in written stories; in spoken French, you would use the passé composé and say ont pleuré.

Why does épousa end in just -a?

This is also the passé simple! For the third-person singular (he/she/it), regular -er verbs like épouser (to marry) take the ending -a. So elle épousa means she married. In everyday spoken French, you would hear elle a épousé.

Could I use se maria avec instead of épousa here?

Yes, both are correct but their grammar differs slightly. Épouser means to marry someone and connects directly to the person (épousa le roi). Se marier means to get married, so you must add the preposition avec (with) to specify the person: se maria avec le roi.

Why is it leur fille and not leurs fille since there are two parents?