Fallow deer come from the Mediterranean area of Europe and the Middle East (Iran and Iraq). They were taken to Europe and parts of North and South America, Southern Africa and Australia, New Zealand mainly as farm or park animals. The parents of these deer came from British Columbia in 1997.n the wild, fallow deer bucks live in herds most of the year. They like each otherâs company. Although they graze all day, they eat most actively at dawn and dusk. In the middle of the day, they rest and chew their cuds.