The phrase "veterinary supervision" means having a veterinarian oversee or be in charge of something, like the care or treatment of animals. Full definition
We recommend antibiotics be used under veterinary supervision where available. (healthforanimals.org)
It is important not to administer these medications without veterinary supervision however since many of them have potentially dangerous side effects. (caninejournal.com)
They should be given only with veterinary supervision and in consultation with someone trained in the use of herbs. (animals.howstuffworks.com)