Good post above by iHeartPieds. I wholeheartedly agree. Spot on post :thumbsup:.
You can try explaining to your parents that this bird is an intelligent, feeling, living, breathing creature. Not low life forms. Therefore, deserves as much to be taken to the vet when injured as any other pet animal would. Just because Budgies don't have a high price tags doesn't mean they don't deserve to be cared for. It is human caretakers complete responsibility to make sure each animal we bring into our home is provided for in a way that makes them as healthy and happy as possible in captivity.
I do hope your parents understand this. Tell them you've talked with experienced adults, and this is what we truly believe.
You can take a pic of the foot if you can, and post it. If it's a fairly common condition such as Bumble Foot for example, like pressure sores on the bottom of foot, we can most likely identify it and what to do as a last-ditch effort, but I can't stress enough that a forum of pet owners is no substitute for an avian vet. Understand that there are so many things it can be, that other pet owners cannot positively diagnose and prescribe a treatment.
Good luck with your parents, and PLEASE keep us posted!