Parallel transport perspective of gauge transformation invariance for connections

Question

Defining a connection on a principal $G$-bundle $P \to M$ is equivalent to defining a parallel transport on $P$ along curves in $M$. With this perspective, Ralph Cohen commented in his notes on the topology of fiber bundles (pp.62) that a connection is fixed by a gauge transformation (gauge group acting on the space of connections by pullback) means the parallel transport defined by that connection commutes with the gauge transformation. Could somebody please clarify for me this interpretation (i.e. gauge transformation invariant means commuting with gauge transformation)?