Oh I'm so sorry, it's definitely Yellow Fungus. Do you know much about Yellow Fungus? I'll be quick and to the point here, because there is only one way to help her, and honestly and truthfully they typically don't do well, although it's more successful when treating an adult.
No more baths, no Neosporin. Yellow Fungus is extremely difficult to treat because it not only grows externally, but also internally. The only treatment known to even give them a chance of survival is a daily oral prescription antifungal medication called Voricozanole in combination with a prescription strength antifungal creme.
Often what happens is people refuse to seek vet care and instead buy an over-the-counter antifungal creme, like Lamisil, and put it on daily. Then either the Lamisil starts to seem like it's helping, or the dragon sheds the areas, and suddenly it looks like it's gone, and stays that way for a week or two...Then BAM it comes back full force, covering the body. This happens because the entire time it is looking better externally it has been aggressively growing internally. Often times they lose entire limbs and tails to it, eyes, etc.
I would stop all baths, as water will only spread it. You can buy an over-the-counter antifungal creme if you want to, but the bottom line is that without long-term, daily, prescription oral antifungal meds (Voricozanole is the only one that seems to have any chance of helping) from an experienced reptile vet, he will not make it, and you have very few if any other options, other than euthanasia. I'm so sorry, but I think it's much better to be truthful and straightforward with you for the sake of the dragon. Especially since your parents think you're "overreacting" and can't afford vet care. Just an FYI, most dragons who survive Yellow Fungus are seeing a certified reptile vet and on the Voricozanole for between 6 months to over a year, that's how tough it is to cure.
I suggest you go into the "Infectious Disease" forum here with your parents, and allow them to read both this post I'm writing and the stories of other people battling Yellow Fungus (also called "Yellow Skin Disease". He does need a skin scraping test to confirm it, but I'm 99.9% sure it's Yellow Fungus that he has, and it's advanced.
Be aware that without starting very aggressive treatment ASAP he could possibly/probably start to lose large chunks of skin/muscle and entire toes/limbs. If this starts then you have to start thinking about whether or not to keep him suffering, for it's extremely painful for them.
I'm extremely sorry this happened to you, I appreciate you wanting to save him, but it was totally unfair of that pet shop employee to only tell you that he had a "fungal infection" and not telling you he actually has "Yellow Fungus" which there is no known cure for and that requires at least many, many months of treatment by a certified reptile vet for them to even stay alive. That's not your fault, and he should be ashamed of himself that he even sold this dragon to anyone, that's actually really upsetting. If the pet shop who had him couldn't afford to treat Yellow Fungus in him then they should have euthanized him.
Also, I guarantee that any reptiles in that pet shop that were anywhere near him or anything he touched also have Yellow Fungus, it is horribly contagious! You need to wash your hands with antibacterial soap before touching him and after touching him, and you also must completely disinfect his tank inside and out and everything in it, including the light fixtures and anything touching or near the tank with very hot water and bleach, or better yet you need to order some F10 Reptile Disinfectant, as it's very hard to kill Yellow Fungus, and if you don't completely disinfect his tank inside and out and everything in it at least once a week he will only keep spreading it all over himself. Use only clean, dry papertowels for his substrate and change them at least once daily. And please make a decision soon about whether or not you can get him started on the oral prescription Voricozanole long-term or whether you need to make a very tough decision, because this will not clear up on its own, it will become extremely difficult to even look at, let alone for him to live with.
Again, this is not your fault, and any pet shop that knowingly sells a dragon who has Yellow Fungus or any other fatal disease that has really no cure (and puts them "on sale" because of it) should be reported and charged the monthly bills for the prescription oral antifungal meds and the Certified Reptile Vet visits.