The question is, as indicated in the title, whether there's a law anywhere in the world that would require parents - explicitly or implicitly - to donate blood (or even organs) to their children if
- They are compatible donors,
- The child's life is in critical danger, and
- The parent will, with reasonable likelihood, not suffer any damage
I would be surprised if there was an explicit law on that. If it could be proved, however, that such a procedure (e.g. drawing blood) would not place any undue burden on the parents, I wonder whether it would fall under some form of an obligation to care for your child.
I also doubt that there's a lot of court cases out there where the parents outright refused and got sued, but I don't know whether there is something out there.
We are of course operating under the assumption that the child is a minor so that the parents are legally obliged to ensure their well-being.
Even if there is no definite answer out there, substantiated opinions are appreciated.