In the mathematics literature, this example is a basic one for SYZ mirror symmetry, going back at least to Gross' seminal paper

http://arxiv.org/pdf/math/0012002v2.pdf

See Chan,Leung,Lau JDG 2012 for a treatment fully within the realm of symplectic geometry. There were also contributions in Seidel and Thomas from the point of view of homological mirror symmetry in http://arxiv.org/abs/math/0001043. 

Edit: Long winded version: One thing to say about your follow up question is that you could say that if we consider the mirror to a nodal variety, there is no reason to expect that it should be the same. So, heuristically, what we expect is that if we degenerate a variety (say to a nodal hypersurface in a toric variety) then the mirror to the nodal variety should have the same number of nodes and that the mirror to the original smooth variety should be given by resolving. Conversely, if we smooth out this mirror, the smoothing will be mirror to some small resolution of the original nodal toric variety. 

 Short version: The fact that the mirror of the resolved conifold is the mirror of the deformed conifold is accidental, purely caused by the fact that the node is,in a sense,
self mirror.