Let $\pi:P\to M$ be a smooth principal $G$-bundle over a smooth manifold $M$, where $G$ is a Lie group. Let $\mathcal{H}\subset TP$ be a principal connection on $P$, and suppose that its curvature vanishes identically. Then for every $x\in M$ there exists a path-connected open neighbourhood $U\subset M$ of $x$ such that, for every $a,b\in U$ and every pair of smooth paths $\gamma_0,\gamma_1:[0,1]\to U$ satisfying $\gamma_j(0)=a$ and $\gamma_j(1)=b$, the parallel transport maps determined by $\mathcal{H}$ satisfy

\begin{align*} \tau_{\gamma_0}=\tau_{\gamma_1}:P_a\to P_b. \end{align*}