Do the parents sin if they keep their daughter’s name “Rama” without changing it, especially since they intend the good meaning of the name, and there is no fatwa prohibiting it?

There is no harm in being named (Rama) unless it is proven to be the name of a temple where other than Allah is worshipped. If that is proven, then it is not permissible to be named with it; because it would be an exaltation of that temple and a promotion of its mention. Furthermore, scholars have agreed on the prohibition of being named with foreign names specific to disbelievers, and the names of idols worshipped besides Allah. Since the name "Rama" is considered a popular hero in Indonesian theater and literature, and a sacred idol among the Balinese; this name must be changed. If official change is difficult, then the person must be addressed by another permissible name.