A commenter asked this as a rhetorical question, suggesting, I think, that the answer must obviously be "no." But I don't see why, assuming that we're talking about a minor child of the parent. Parents are rightly seen as having duties to their children. These include the duties to work to support the child for 18 years (more controversially, that's extended even beyond 18 years in many child support decisions, but for now I set that aside); to care for the child; to bear a post-viability fetus, at least absent some substantial threat to the mother's life or health; and more.
Why wouldn't this also extend to the obligation to provide a life-saving transplant, at least when the risk is as low (not zero, but very low) as it is for kidney transplants? You bring a child into the world, and you incur major obligations to it; why shouldn't this be one of them?
Now I don't ask these questions as rhetorical ones, since it's possible that some distinctions can be drawn (or even that the existing obligations on parents are excessive, though I'm skeptical about that). Perhaps there is a dispositive difference between providing an organ and having to work for 18 years to support someone. (I agree there's a difference, even an important one, but it's just not clear to me that it should lead to a difference in result.) Perhaps there is something dispositively different between having to give a kidney forever, and having to provide one's womb for several months; or perhaps women shouldn't have to bear fetuses even post-viability; or perhaps women only have to bear fetuses post-viability because they knew of this obligation early on, and had an opportunity to avoid this by a pre-viability abortion. I haven't thought about the matter deeply enough to have a well-worked out response to all these things.
But my intuition is that a legal duty to provide a kidney, given the very low risk that it involves, is well within the range of burdens that parents may rightly be required to bear; and at the very least we can't just categorically exclude that possibility. I'd love to hear what others have to say.