Oh dear! I really feel for you that you are stressed and worried about your piggies and other things right now. I really encourage you to raise all this with your parents though, as between you there are some important things to consider and decisions to make which is a discussion that can't really be avoided. And also, it may not even turn out as bad as you are thinking right now
Your piggie does need to see a vet soon, and seeing the regular vet that your parents like to use is still far better than trying to treat this yourself. Even if you don't feel they are as in tune with piggies as you'd like, they are still qualified vets with medical training that have probably treated lots of bite wounds before.
And about the re-homing idea; this is a tough one. Because sometimes if it doesn't work out you need to look rationally at what is the best thing to do next.
After my piggies had a fight I had to seperate them. I am an adult with a full-time job, but also only room for 1 cage. And guinea pigs are sociable animals who do not like to live by themself. So I had to take this baby piggie who had bitten my adult piggie back to the Rescue Centre where he came from.
I cried all the way there, but i don't regret doing it. Because obviously he was never going to be friends with the piggies I already had. And i had no space for him to keep him. And it wasn't easy, but sometimes you have to be practical. Though it hurt at first for me to take him back, now i look back at his photo and feel happy that the Rescue centre I returned him to managed to find him a friend that he didn't want to bite and he then went to a lovely new home. That is a much better outcome for him than keeping him in a cage that's too small where he will be by himself all his life. Maybe deep down you can see that asking a rescue for help to rehome him might actually be the best outcome (and i'm so sorry that's not easy to hear, sometimes it is just kinder to make that choice).
I really am sorry you're going through this, it wasn't easy for me when I had this piggie problem. But please do talk and listen to your parents so that together you can act the best way forward.
Best wishes & big hugs to your piggies and family
