I don't think it is related to isometry.
In a curved spacetime, there is no obvious notion of "parallel". But you need one, so you just define it, by defining the notion of a derivative, a rate of change, an acceleration. No acceleration = no change in velocity = no change in tangent vector = parallelly transported (by definition, but hopefully it will seem like a graceful generalization of the terms we use for flat spacetime). There are many possible dervatives, and hence many possible notions of parallel. In GR, the derivative is chosen by specifying that the connection be the Levi-Civita connection http://en.wikipedia.org/wiki/Connect...mathematics%29