In some countries, they are. In other countries, people don't even need a prescription to obtain them.
In the UK, a medically qualified doctor or veterinarian must prescribe antibiotics for the general public, pet owners or farmers to be able to obtain them. In farming, the vet has a responsibility to check whether antibiotics are required and the health of the animals involved. In terms of administering antibiotics to any animal, this can be carried out by pet owners or farmers under the direction of the veterinarian. In asking whether antibiotics should be administered only by a vet, this has not been deemed necessary in the UK. However, undertaking additional training on administration of antibiotics is increasingly a requirement under some farm assurance schemes or supply contracts.