I understand the definition of parallel transport given the data $(M,E,\nabla,\gamma)$ and I understand the definition of geodesics given the data $(M,g)$. Given all of this data, we can make the following definitions:
Fix some $y\in M$ and consider a normal neighborhood $U$. Given some $x\in U$, we can consider the $\mathbf x\in T_yM$ defined by $\exp_y(\mathbf x)=x$ and in addition we can consider the geodesic $\gamma_x:[0,1]\to M$ defined by $\gamma_x(t)=\exp_y(t\mathbf x)$. Furthermore, for each $v\in E_y$ we can consider $$\tau_v(x):=V(1)\in E_x,$$ where $V\in\Gamma([0,1],\gamma_x^*E)$ is the parallel transport of $v$ along $\gamma_x$.
Why does this solve $\nabla_{r\partial_r}\tau_v=0$ (as claimed in the proof of Theorem 2.26 in Heat Kernels and Dirac Operators)? Roughly speaking we have that $\nabla_{\dot{\gamma}_x(1)}\tau_v=0$ (by the definition of parallel transport), so my guess would be that there is some correlation between the velocity of $\gamma$ and the radial vector field $\partial_r$. Can someone elaborate?