The belief that horsehair can paralyze animals is an old wives' tale. There is no scientific evidence to support this claim. The likely origin of this myth is that horsehair worms, which are parasitic worms that can grow quite large, can sometimes be mistaken for horsehair. These worms can be found in insects, and they can cause paralysis in their hosts, but they are not directly harmful to birds like pheasants.
If you are ever concerned about the health of your pheasant, it is best to consult with a veterinarian.