Let $M=\mathbb{S}^2\setminus\{x_S\}$ be the sphere without the South Pole $x_S$, endowed with the classical metric $$g=\begin{bmatrix}1 & 0 \\ 0 & \sin^2(\theta)\end{bmatrix}.$$ For any $x\in M$ let $\gamma_x$ be the unique geodesic from $x$ to the North Pole $x_N$. I have to compute explicitly the parallel transport of any vector $v\in T_xM$ from $x$ to the North Pole.
My attempt: by linearity it suffices to compute it for the basis $(\frac{\partial}{\partial\theta},\frac{\partial}{\partial\varphi})$. Letting $\{e_1,e_2\}$ be the canonical basis of $T_{x_N}M$ endowed by the inclusion $\mathbb{S}^2\subset \mathbb{R}^3$ we can actually claim that for any $x=(\theta,\varphi)\in M$, it holds $$P_{\gamma_x}(\frac{\partial}{\partial\theta}\Bigl\vert_{(\theta,\varphi)})=\cos(\varphi)e_1+\sin(\varphi)e_2,\quad P_{\gamma_x}(\frac{\partial}{\partial\varphi}\Bigl\vert_{(\theta,\varphi)})=\sin(\theta)\Bigl(-\sin(\varphi)e_1+\cos(\varphi)e_2\Bigr),$$ where $P_{\gamma_x}$ is the parallel transport to the North Pole along $\gamma_x$.
Does it looks correct? I found this result only looking at the problem geometrically (imagining the vectors moving) and also knowing that the parallel transport preserves the inner product (that is why $\frac{\partial}{\partial\varphi}$ is rescaled).