A smooth vector field v along a parameterized curve γ(t) in a surface X is parallel if the covariant derivative Dv/dt is zero. For a plane X, this means that v is constant. γ(t) is a geodesic if and only if the tangent vectors γ′(t) are parallel.
If v is parallel and t1 < t2, then γ(t2) is the parallel transport of γ(t1) along γ.