No, a differentiable manifold is not enough to have geodesics. You also need a metric on the manifold, which makes it a Riemannian (or pseudo-Riemannian) manifold.
Your definition involving
Also let the tangents to the cisedoeg all be parallel.
doesn't really make sense because manifolds (Riemannian or otherwise) do not come with a canonical identification of the tangent spaces that you can use to define whether two tangents at different points are parallel.
In a Riemannian manifold you can require that the tangents to the curve all move into each other when "parallel transported" along the curve. However, this turns out to be an alternative characterization of geodesics, not some kind of anti-geodesic.
I suspect the source of your problem is that your basic premise
If I understand correctly, a geodesic between two points a and b is the "most direct" path from a to b.
is not completely correct. A geodesic is a curve that is locally the shortest curve between two points on it -- where "locally" means that it only means to be shortest between two suffiently close points on the geodesic.
In particular a great-circle arc on a sphere that is longer than 180° is still a geodesic. (An entire great circle, traversed again and again as the parameter increases to $\infty$, is a geodesic).