Lots of parasites are not host specific and certain ones can actually lead to death if an animal eats that prey. However dangerous parasites are generally rare or only found in certain animals. The ones that infest many different animals your animals have already been exposed to. Like I said the problem only comes if there is a high concentration of them. Keeping lots of birds in a given space will be even more likely to get you a parasite problem as feeding raw game and pests like mice but the former won't have nutrition and health benefits.