This case is not a proper one to understand the parallel transport and the role of the Christoffel Symbols. This is due to the fact that the latter are all identically zero.
For the parallel transport of a vector $\:V^\alpha\:$ along a curve with parametric equation \begin{equation} \boldsymbol{p}\left(\lambda\right) \boldsymbol{=}\left[p^1 \left(\lambda\right), p^2 \left(\lambda\right)\right]\qquad \left(p^1\boldsymbol{\equiv}p,\:p^2\boldsymbol{\equiv}q\right) \tag{01}\label{01} \end{equation} we use the equation \begin{equation} \frac{\mathrm dV^\alpha}{\mathrm d\lambda} \boldsymbol{=} \boldsymbol{-}\Gamma_{\beta\nu}^{\alpha}V^\nu\frac{\mathrm dp^{\,\beta}}{\mathrm d\lambda} \qquad \left(\alpha,\beta,\nu \boldsymbol{=}1,2\right) \tag{02}\label{02} \end{equation} or equivalently \begin{equation} \mathrm dV^\alpha \boldsymbol{=} \boldsymbol{-}\Gamma_{\beta\nu}^{\alpha}V^\nu \mathrm dp^{\,\beta} \qquad \left(\alpha,\beta,\nu \boldsymbol{=}1,2\right) \tag{03}\label{03} \end{equation} The <PERSON> symbols are expressed through the components of the metric tensor $\mathbf{g}$ \begin{equation} \Gamma_{\beta\nu}^{\alpha}\boldsymbol{=} \frac{1}{2}g^{\alpha k} \left(\frac{\partial g_{k\nu}}{\partial p^{\,\beta }}\boldsymbol{+}\frac{\partial g_{\beta k}}{\partial p^{\,\nu}}\boldsymbol{-}\frac{\partial g_{\beta \nu}}{\partial p^{\,k}}\right) \tag{04}\label{04} \end{equation} In all equations we make use of the Einstein summation convention. 616