He was the second son of Bernard Rochford and his wife Angela Teresa Kelly. He had two younger brothers and a sister. Bernard Rochford was Called to the Bar of the Inner Temple on 20 June 1912. He married the following year. For many years, the Rochfords lived at Rosemount, Sunningdale, Berkshire.

Bernard’s father, Joseph, had started a business growing fruit under glass. He erected his first greenhouse in 1872 and was the founder of what has become in the 21st century, Rochfords Nursery and is one of the largest wholesale suppliers of hardy plants in the United Kingdom. When he died in 1932, he left £278,692-6s.

Anthony was educated at Ampleforth, as was his father. [Ampleforth claimed that there was barely a year in the 20th century that there had not been a Rochford at the school.]  He Matriculated in 1938 and spent a year at Christ Church.

He was gazetted to the 1st Battalion Irish Guards on 14 January 1941 and played for the Irish Guards Rugby team in the 1941-2 season.

There are conflicting reports of the action on 30 March 1943, but it is known that three men were killed, twelve wounded and over a hundred were missing – some of these were taken prisoner, along with several hundred more.

Anthony was reported missing and confirmed killed at the end of October.

He is commemorated on the Medjez el Bab Memorial Face 13.

Anthony was a first cousin of Stephen Rochford. Their fathers were brothers.