I have seen many claims that parallel transport in thin homotopy invariant, but I cannot seem to prove it. My conventions are:
Let $\pi\colon E\to B$ be a principle $G$ bundle with connection $\omega$. Then we can define the parallel transport along a smooth curve $\gamma\colon I\to B$ by $\Gamma_\gamma\left(u\right)=\gamma^\mathrm{h}_u\left(1\right)$ (where $\gamma^\mathrm{h}_u$ is the unique horizontal lift of $\gamma$ starting at $u$).
Let $\gamma_0,\gamma_1$ be smooth curves with same endpoints. and $H\colon I\times I\to B$ be a homotopy (endpoint preserving) which is smooth. Call it thin if $\mathrm dH\vert_{\left(s,t\right)}$ does not have full rank. In this case, $\gamma_0$ and $\gamma_1$ are thin homotopic.
I need to show that if $\gamma_0,\gamma_1$ are thin homotopic, then $\Gamma_{\gamma_0}=\Gamma_{\gamma_1}$.