Nicknamed "Ol' Spaghetti Legs", Parker became one of the most important parts of the Eskimos dynasty of the mid-1950s. While Parker played both defence and offence, a not uncommon practice in his day, his main strength was offense, where he was both a strong runner and an excellent quarterback. This is shown by his Western All-Star selections; as a running back in 1954, 1957 and 1959, and as a quarterback in 1955, 1956, 1958, 1960 and 1961. Parker won six consecutive Jeff Nicklin Memorial Trophies as the West's most outstanding player from 1956 to 1961, and seven in total. He was awarded the Schenley Award as Canadian football's most outstanding player in 1957, 1958 and 1960 and was the runner-up in 1956 and 1961.