Parker was born at Tredington (which was then in Worcestershire). His father, a clergyman, was a son of Sir Henry Parker. His paternal grandfather had married a daughter of Alexander Hyde, Bishop of Salisbury. He began his career at sea in the merchant service. Entering the Royal Navy at the age of 24, he was made lieutenant in 1744, and in 1748 he was made post-captain. In his royal navy career, he captured a Spanish galleon that was worth £600,000. This gave his family its wealth. Currently, his descendants live in the south wing of Melford Hall.