Only the form of payment would be the issue. If they use the parents Credit Card or check book then yes, they need parental permission. If they find cash and pay for it that way, then all the power to 'em. I'd start the project upon receipt of the down payment. However, I'm a hobbyist and not a pro-level advertised fursuit maker. So it would just be an agreement between friends. Not a business transaction. A Pro-level maker, like Bettlecat and others I'm sure would have to seek permission from the guardian/parent before starting. It's their business and why jeopardize your primary income just to give some kid a fursuit?
So there are a lot of variables in your situation. I don't see how one answer would fit all the bills.