Imagine you're standing on the equator facing north. Now imagine you start walking. You keep walking in the same direction until you get to the North Pole. Now, without changing the direction you're facing, you start to strafe to the right. Keep walking this way and you'll eventually get to the equator again, a quarter of the way around from where you started from. Now, again without changing the direction you're facing, walk backwards. You'll eventually get pack to the point you started. However, you're now facing a different direction from when you started, even though you kept facing the same direction at every step of your journey. This is because the Earth is a curved surface.
Every point on a manifold has its own linear space of tangent vectors (called the tangent space). In order to be able to talk about two tangent vectors at different points being parallel, we need an additional structure called a connection which enables us to relate vectors in the tangent spaces of points on a path to one another as we follow the path. Two tangent vectors at different points might be considered parallel if one path between the two points is followed but not parallel along another path.