Is it obligatory to wear a khimar that covers the back and shoulders, or is it sufficient to cover the chest with loose clothing, taking into account parental opposition and the general social situation of girls?

The obligatory extent of a woman's hijab is achieved by any garment that covers her awrah (private parts) without describing or revealing it. It is not a condition that the shoulders or back be covered by the head covering itself, nor is a specific color or design required. Rather, the requirement is that it should not be eye-catching. As for the "clothed, yet naked" woman, she is one who wears something that covers her awrah but describes or reveals it, or she wears something that exposes a part of her awrah.