Assume $(U,x)$ is a coordinate with $\{\frac{\partial}{\partial x_k} \}$ is orthonormal basis. Let $c=(c_1(t),c_2(t))\in M^2$ and $V=V^i \frac{\partial }{\partial x_i}$. What I do :
(a) since $V$ is parallel, we have $$ 0= \frac{DV}{dt}= (\frac{dV^k}{dt}+ \frac{dc_i}{dt}V^j\Gamma_{ij}^k )\frac{\partial}{\partial x_k} $$ So, there is $$ \langle \frac{dV}{dt}, \frac{\partial}{\partial x_k} \rangle = - \frac{dc_i}{dt} V^j \Gamma_{ij}^k $$ for normal coordinates, we have $\Gamma_{ij}^k = 0$ ? (I am not sure). So, we have $$ \langle \frac{dV}{dt}, \frac{\partial}{\partial x_k} \rangle = 0 $$ if the above process is right, obviously, it depends the coordinate. And for general coordinate, the above process is not right.
(b) Consider parametrized $$ \gamma : \theta \rightarrow S^2 ~~~\theta\rightarrow (\cos\theta, \sin\theta, 0) $$ So, the velocity is $$ V=(-\sin\theta, \cos \theta, 0) $$ Then to calculate $\frac{DV}{dt}$. I think I should choice a suitable coordinate on $S^2$. But
