My current understanding of parallel transport is the following:
Let $P \xrightarrow{\pi} M$ be a principal $G$-bundle and $\gamma: [a,b] \rightarrow M$ a curve in $M$. For a given connection $A$, we may define a map $$\Pi_\gamma^A: P_{\gamma(a)} \rightarrow P_{\gamma(b)}\\ p \mapsto \gamma_p^*(b)$$ where $\gamma^*_p$ is the horizontal lift of $\gamma$ such that $\gamma^*_p(a) = p$.
Thus we see that parallel transport is a way to map a point in one fiber to another, hence it maps points in $P$ to points in $P$.
What has been confusing me is that many resources, such as Wikipedia or Wolfram MathWorld, instead describe parallel transport as transporting vectors. I do not see how the above definition of parallel transport has anything to do with vectors unless we consider $(\Pi_\gamma^A)_*$, the pushforward of the parallel transport map, but this is never mentioned.
What is the connection between these two viewpoints?