In books I read, the notion of parallel transport along a curve on manifold is defined in terms of connection. Then we have a theorem, which states, that we can recover connection given parallel transport.
So, the question is whether there is a simple enough definition of parallel transport, that does not depend on connection. Especially interesting is the case of riemannian manifold.
I think the question is important, because parallel transport is fundamental concept, which can provide helpful insight towards understanding connections. Of course, that is if we can properly define it.