What would be a reference for the following property of symmetric spaces?

Given a smooth curve $\gamma : [0,1] \rightarrow M$ on a symmetric space $(M,g)$, there exists an isometry $\varphi : M \rightarrow M$ taking $\gamma(0)$ to $\gamma(1)$ and such that the parallel transport map $$ P_\gamma : T_{\gamma(0)}M \longrightarrow T_{\gamma(1)}M $$ equals the differential of $\varphi$ at the point $\gamma(0)$.

I guess one can get this from two particular cases: (1) the curve $\gamma$ is closed and we are looking at holonomy vs. isotropy groups, and (2) the curve $\gamma$ is a geodesic segment and is the projection of a translation of one-parameter subgroup of isometries, but I would love just to be able to cite something.

I looked up in Helgason and it does not seem to be there, at least in this guise.