The earliest recorded use of the phrase 'sick as a dog' is dated 1705. At much the same time, people were using other phrases such as 'sick as a horse' and even - believe it or not - 'sick as a cushion'! More recently, we seem to have discovered, especially if we happen to be football-players, the phrase, 'sick as a parrot'. It is all just part of the rich tapestry of English usage.