Quote by Anamitra
So if we parallel transport a vector along different paths starting from the same point the components do not change,when referred to the local inertial frames.

This is really a circular argument. At any event in spacetime there isn't a single "locally inertial" coordinate system. You have a choice: for a start, you can rotate the spatial axes and get an equally good locally inertial coordinate system. Also any coordinate system moving at a constant velocity to a locally inertial system is locally inertial.
Given that choice, how would you choose which family of locally inertial coordinate systems to use along a worldline? The answer is you'd use parallel transport!
If you have two different geodesics (or any other worldline) joining a pair of events, if you paralleltransport a locally inertial coordinate system from one event to the other, you could get two different locally inertial coordinate systems at the other end!