I think that with good communications that we have today, a parent should be with a child that needs treatment. If the condition is really critical, a written permission for treatment from a parent is sufficient, right? But, I think, it's still the obligation of a parent to be with a sick child. Thanks for the question.

If you think narrowly only in terms of standard medical ethics (let's leave social philosophy aside for a moment) then I would say that parents are default decision-maker as it is assumed they know best their child and they need to handle the consequences of the decisions. But they are also only a default decision-maker. If need be someone else can be appointed and they should, just like the parents, follow best-interests standard in medical decision making.

One thing that strikes me is that children are an exception among incompetent patients. This can vary in different jurisdictions, but generally there is not such strong presumption of family knows best with other types of incompetent patients. And of course there are many examples where parents do not really know (or don't act on that knowledge) best what are the child interests e.g. due to their limited understanding of the problem, or one or other type of predjudice (FGM comes to mind, or my personal "favorite" parents in antivaccination / alternative "medicine" movements).

Now if you think of it in wider terms there might be such rights, but they are probably only prima facie rights and subject to limitation by what is practicable. Unfortunately funds for such procedures carried out in far-away countries are always below what is needed and additional travel costs may in some cases be out of reach. This is probably not "right" in some very general sense, but it is not "right" that children get sick and die in the same sense. It is a question of what we can do in this "unjust" world to make things slightly less "wrong".

I think this is really good topic for further analysis and a global bioethics paper.