"František and Žofie live at her parents'."
Translation:František a Žofie žijí u jejích rodičů.
You have to decline the possessive pronoun. The correct form of její 'her' is in this case jejích (masculine plural, genitive case) The pronoun and the noun it refers to (rodiče) have to match. (same case, same gender, same number...)
'Their' is jejich. It's not declined. Note the difference in vowel length.
Rodiče should feel rather plural. Then, the -ič ending is chiefly masculine and denotes an actor of some process, like the Latin -tor, english -er. Řidič, vodič, vařič. The feminine complement is -ička.
rodit is a verb and means to give birth. There is also feminine rodička, but that is a woman actually about to give birth and soon after giving birth.
Do not get confused with the -ice ending, which is indeed feminine. podvodník/podvodnice, služebník/služebnice, slepice, krabice, opice.
The -ička can also a dimunitive of these: slepička, krabička, opička. However, there is no rodice (there was rodicě in Old Czech).