I am having trouble understanding the "meaning" of the one of the ways to get the geodesic equation in GR. The first way, which seems to me also the most intuitive, is to say that the geodesic is the path with extremise the GR "action" (if I may call it that, it looks like a distance though):
$$S = \int(\epsilon g\_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda})^{1/2}d\lambda$$
Where $\epsilon$ is -1 or 1 for time-like and space-like paths respectively. I am not sure about the case of light-like paths.

I say it is intuitive, because the method closely resemble the way we use to minimize the length of a path on a surface, i.e. we minimize $L = \int ds$.

However, there is also another way to obtain the geodesic equation, and that is to say that the geodesic "path" is the one which parallel transport it's own tangent vector. I am struggling to understand this definition. I know the equations gives the right result, but I'd like to get an intuitive understanding.

Indeed, I think it has something to do with the fact that the parallel transport path is extremal, so that if the tangent vector follows it, it's a geodesic. 

I don't know if that's clear, so I'll try to explain it with an analogy. In Euclidean space, the gradient points to the direction of greatest increase of the function. So, given two values of the function, the "fastest" way is to follow the direction of the gradient. Is there a similar interpretation for the definition of the geodesic ?