Verify that parallel transport on $S^2$ depends on the curve.
The first curve is made of two geodesics: a geodesic $\g_1$ joining the north pole $N$ and a point $P$ on the equator followed by a half circle $\g_2$ around the equator. This curve ends at the point $Q=-P$. The second curve is the geodesic $c$ joining $N$ and $Q$. Let $T$ be the tangent of $\g_1$ at $N$. The parallel transport of $T$ along $\g_1$ to the point $P$ is the tangent $T_1$ of $\g_1$ at $P$. As $T_1$ is pointing to the south pole $S$, the parallel transport of $T_1$ along $\g_2$ to the point $Q$ is a tangent vector $T_2$ at $Q$ pointing to the south pole. On the other hand the parallel transport $C$ of $T$ form $N$ to $Q$ along $c$ is pointing to the north pole, i.e. $C=-T_2$.