Given a manifold with affine connection $(M,\nabla)$ and a curve $\gamma:I\rightarrow M$, let $$T_{\gamma}^{t_0,t_1}:T_{\gamma(t_0)}M\rightarrow T_{\gamma(t_1)}M$$ be the parallel transport over $\gamma$, for each $t_0,t_1\in I$. We know such map is defined by means of the unique solution of $D_t V=0$ under the initial condition $V(t_0)=V_0$, making $$T_{\gamma}^{t_0,t_1}(V_0)=V(t_1),$$ where $V(t)\in T_{\gamma(t)} M$ for every $t\in I$ and $D_t$ is the covariant derivative along $\gamma$.
Now for the question: suppose $\gamma$ is a closed curve, say $\gamma(t_0)=\gamma(t_1)=p$, and let $\gamma'(t_0)=X_p$ and $\gamma'(t_1)=Y_p$. If $\widetilde{\gamma}$ is another curve with the same "initial conditions" ($\widetilde{\gamma}(t_0)=\widetilde{\gamma}(t_1)=p$, $\widetilde{\gamma}'(t_0)=X_p$, $\widetilde{\gamma}'(t_1)=Y_p$), does it follow that $$T^{t_0,t_1}_{\gamma}=T^{t_0,t_1}_{\widetilde{\gamma}}?$$ I don't know much of holonomy (which I eventually stumbled on when looking for this problem), but any help would be of value.