I'm so, so sorry...
He didn't have a "cough", he died from a GI infection that started inside of his crop, because the breeder sold you an unweaned baby, and you didn't know that his formula had to be between 104-110 degrees F, nor did you know that he had to be kept inside a Brooder that needed to be around 85 degrees...If the vet you took him to didn't mention that to you, then he wasn't an Avian Vet, at least if you gave him all of the information...
Did that vet do a culture? As I said, he most-likely have a Fungal Infection, which is usually what happens, and also maybe a Bacterial Infection, but without a culture he wouldn't have known...and the Antibiotics would have only made a Fungal infection worse...I'm so sorry...This happens at least once a week on this forum to someone who buys an unweaned baby parrot but is given no information about formula temperature or ambient temperature the bird must be kept at...he died from an infection in his crop, which started because you can't feed a baby bird "warm" formula...it must be at least 104 degrees F and no hotter than 110, this is an extremely precise process and there is no room for error, obviously, so if you just fed him "warm" formula, as the breeder told you, the formula simply sat inside his crop and spoiled, causing a Fungal infection and possibly also a Bacterial infection...
****You need to call that breeder and tell them that he gave you horrible advice, that he didn't tell you that the formula had to be between 104-110 degrees F, that you didn't know he had to be kept in a Brooder/at 85 degrees , and now he has died from a GI infection due to spoiled formula in his crop...
***Also, please, please do not ever buy another unweaned baby bird that you have to feed formula to or have to keep in a Brooder, chances are it will only happen again...I bred parrots for over 20 years, and any breeder who is willing to sell unweaned baby birds, let alone not even give you this very basic information to which the baby's life depends upon, is a horrible, unscrupulous breeder who only cares about making money...
EDIT: By the way, also, don't take another bird back to that same vet either...birds don't get "coughs", coughing is only a symptom that birds will display when they have either a Bacterial or Fungal Infection, and a Culture/Swab must be taken to diagnose exactly what microbe is causing the infection, which will tell them exactly what medication is needed...If that Vet didn't take any type of Culture but simply just said "he has a cough" and gave him Baytril (Enrofloaxcin), the he's not a Certified Avian Vet, he may be an "exotics" vet, but that doesn't mean anything at all, and he doesn't know anything about birds...I can't believe he didn't explain the formula/ambient temperature problem to you like I did...that's awful too...I'm so sorry that you went through this...Just from now on, please only buy a totally weaned baby bird that doesn't require you to feed it formula or anything special...Hand-feeding a baby bird will NOT make the bird bond more-closely to you, that's a myth...All you need to do is find a responsible breeder who cares about their birds and not only about money, and who sells fully-weaned, hand-raised babies...that's it, never buy another unweaned baby bird, it's too dangerous and there are too many variables to what you have to do, and if you don't have prior education or experience then it's extremely difficult to finish successfully...But it's not your fault, it's that horrible, unethical breeder's fault...