Wow, that's an awful thing, especially the uneducated and inexperienced vet, that statement about "respiratory infection in juveniles being very rare" might be the dumbest thing I've ever heard a vet say. She may have been catering to your parents wishes, like they told her to say the beardie was fine, or she really is an uneducated vet that should not be treating reptiles at all. Either way unfortunately the only ones that suffer due to her ineptness are your beardie and you, because those are all classic signs of a severe upper respiratory infection, and unfortunately the only way to help your beardie is to find a qualified, experienced reptile vet, otherwise your beardie will not survive much longer.
If you did already find a vet that you believe is qualified and willing to help your beardie, please do a couple of things. First, ask as many questions as you can. Make this vet do the proper things, like taking a culture of your beardie's mouth/nostrils/throat and sending it out for testing to ensure that he gets the correct antibiotics. And please do not leave this vet without a full course of antibiotics, because this is the only way he's going to get well. They may call you later on after the culture comes back and tell you to switch antibiotics, if they do then please do it because the culture will tell them exactly what bug(s) he has and what medications will treat them. They will likely put him on a broad-spectrum antibiotic and maybe an antifungal as well, because with upper respiratory infections in Beardies they often get a combination of a bacterial infection and a fungal infection, like aspergillus, so a lot of the time they will put them on both an antibiotic and an antifungal at the same time, which is OK. But after the culture comes back from the lab they might have to switch medications based on the results, so make sure that is all done.
Also, please ask for probiotics to give your beardie along with the antibiotic and antifungal medications, because these medications are very tough on their gastrointestinal systems and they often stop eating and have bowel movement issues like runny poop due to the meds. If you give him probiotics during the treatments they typically do a really good job of making their tummy feel better and increasing their appetites.
Also keep in mind that reptiles, being cold blooded, heal very, very slowly, and it often takes a very long course of antibiotics and antifungals, sometimes multiple courses, to get rid of the Infections (another reason to ask for probiotics, if the vet tells you that probiotics aren't necessary tell him you want them anyway). Treating a beardie upper respiratory infection take a lot of time and patience, but it sounds like you're ready and willing. I appreciate you spending your own money to help your beardie, most teenagers would not do that. But as I said, he has all the classic symptoms of a severe upper respiratory infection and he's also very uncomfortable and in pain, as represented by the
black beard, and it's 100% necessary to get him a culture and immediately get him on proper prescription medications, because he will not survive much longer without them.
And please make sure you get copies of everything from the vet, all of the paperwork that states his vitals like his weight, and all of the notes about their physical findings and diagnosis and what medications were given along with detailed dosing instructions, and also be sure to get a copy of any tests they run, and when the results come back and they call you with them make sure you either go pick up a copy of the results or have them email them to you, because it's not unusual for vets to not want to give paperwork...
I hope the vet you found is a qualified, experienced Reptile vet with bearded dragon experience, and I hope they know what they're doing this time. I suggest that after you get a diagnosis and he's put on medications (and especially after you get the culture results back) that you take a copy to that first vet and tell her that she has no business treating bearded dragons, and that upper respiratory infections in Beardies, regardless of age, are one of the most common illnesses they can suffer from. We see upper respiratory infections in baby Beardies, juveniles, and adults on here every day, and she needs to know that if a bearded dragon has a constant
black beard, that means the they are uncomfortable and in pain, and if she doesn't even know that then she shouldn't be seeing them. That really is frustrating. And I hope your parents also learn from this experience too after he gets diagnosed and put on multiple medications. I'm sure your parents mean well, but they really need to realize that finding a veterinarian that is experienced in treating reptiles and who knows what they're doing is very rare, and that it's the people like us, in the online forums, who know what we're talking about because we have spent years and in some cases our entire lives raising and caring for bearded dragons. The same goes for people in bird forums, fish forums, ferret forums, etc. Vets go to veterinary school for 4 years and they can specialize in a specific species if they want to AFTER they get their general veterinary training. It's exactly the same as doctors that see people, I'm sure your parents wouldn't go see their general practitioner or family doctor to treat a broken bone, they'd go to an orthopedic surgeon. It's the same thing, only the number of vets that specialize in reptiles is very low....
Good luck, and I hope you get your beardie to the vet soon. You should bump up his temperatures in the meantime by 5 degrees or so, that can help a bit, and make sure you remove any water dishes from his enclosure. And no baths until he sees the vet either.