Quote by Phrak
hmm. what would be the result of parallel transporting a covariant vector?

Using the abstract approach, you can parallel transport an arbitrary tensor along a curve. If you have some tensor [itex]T^{qwer}{}_{uiop}[/itex] you parallel transport it along a curve with tangent t^a by insisting that along the curve
[itex]t^a \, \nabla_a T^{qwer}{}_{uiop} = 0[/itex]
Though as I said, the intuitive meaning, if any, tends to get lost along the way, at least for me.