Thank you for the parable. I agree that it is a helpful way to discuss the issue, especially for scientifically-challenged people like me. Though I do not consider myself a theologian, that is definitely the side from which I approach the issue. Allow me to answer the question which you pose at the end of the parable, which I will here paste:
My question to the theologians: Why might God choose not to leave evidence that this 100-year old tree is only a week old?
I think He did leave evidence. The question is: "Where is that evidence to be found?"
Auto manufacturers routinely install a convex mirror on the outside of the passenger door to give a driver the widest possible field of view for other cars that might be beside and behind him. Engraved on that mirror are the words ""objects in (the) mirror are closer than they appear." Why would the manufacturer install a mirror that gives an appearance not fully aligned with reality? Because it's more important for the driver to be aware of all the vehicles in that area than their individual exact distance from him. As is so often the case in life, it's a matter of trade-offs. The advantages and disadvantages of a convex mirror in that situation are overall more desirable than those of a regular mirror. Where is the manufacturer's warning to the driver that the representation he is seeing does not fully represent reality? It is not in the mirror, for that would compromise the mirror. Rather, it is on the mirror, in the words the manufacturer has engraved.
Likewise, God has given evidence that the 100-ring tree is only a week old, but He has not put that evidence in the tree. Rather it is in the words that God has sent - in the case of the parable, through the theologian. Moreover, it is as if the words have been put on a sign staked in front of the tree - for both the scientist and theologian to read. That is, modern science, as I understand it, was birthed, and has grown best, in cultures aware of the Bible. In other words, the staked sign has never been far from the tree. We can reasonably consider it to be like words engraved on the mirror.
Let me hasten to add that the example of the convex mirror does not prove that the earth is younger than it appears. Nor am I saying that this is the only kind of motivation God could have had for doing such a thing, if, in fact, He did it. I am only saying that this strikes me as one reasonable way of thinking about why appearance and reality might legitimately diverge.