I often see a sentence like "the mirror of resolved conifold is the deformed conifold" in physics literature. I would like to ask, in what sense is this true? What is known for mirror symmetry for these manifolds in mathematics?

I am aware of the naive duality of K"ahler parameter (volume of the exceptional $\mathbb{P}^1$) of the former and complex parameter (radius of the vanishing $S^3$) of the latter, but I expect there should be more to say. I would appreciate any good reference to look into.

Edit: I have one more question. Why does this situation globalize? Calabi-Yau 3-fold with conifold singularity may smoothed either by resolution or by deformation of the singularity, they are not mirror each other.