Position/orient parent to place child at a particular position/orientation

If you want a parent world rotation$P_W$ that takes the child local rotation $C_L$ to the provided child world rotation $C_W$, we can express that as an equation:

$$C_W = P_W cdot C_L$$

Then you can simply solve for the parent rotation algebraically:

$$C_W cdot C_L^{-1}= P_W cdot C_L cdot C_L^{-1} = P_W cdot I = P_W$$

So, parent.rotation = childTargetWorldRotation * Quaternion.Inverse(child.localRotation);

(Note that the order of the multiplication here is important - putting the inverse on the left side of the multiplication does not produce the same rotation, in general)

Now that we know the rotation, we can work out how that transforms the parent-to-child vector:

Vector3 worldOffset = parent.TransformVector(child.localPosition);

And then reverse that vector to find where to place the parent:

parent.position = childTargetWorldPosition - worldOffset;