。
His parents tried to calm his feelings.
Literal
Parents [topic-は] his feelings [object-を] calm-try-to did.
~ようとした means 'tried to do X' — the volitional form + とする expresses an attempt. 気持ちを静める (to calm someone's feelings) is more specific than just 静める. The past tense ~ようとした implies the attempt without confirming success.