"They love their parents."
Translation:Они любят своих родителей.
Using их is like saying that they like "their parents" but "their" is meaning another group of people. Своих is closer to saying "their own parents." In English we use the same "their" for both, but it could be confusing for whom "their" is standing in.
Nominative singular is «роди́тель» 'parent', genitive singular is «роди́теля».
Nominative plural is «роди́тели», genitive plural is «роди́телей».
Oh, sorry, it's accusative here of course. For animate plural nouns, accusative is always the same as genitive.