Let $M$ be a Riemannian manifold, $p\in M$ and $X,Y\in T_pM$. Take the points $q = \exp_p(sX)$ and $r = \exp_p(sY)$ and assume that there is a unique geodesic of minimal length from $q$ to $r$ for any $s\in[0,1]$ (we might further assume that $M$ is geodesically complete or even compact to guarantee existance of geodesics). Hence there is $Z_s\in T_qM$ such that $r = \exp_q(Z_s)$ for any $s\in[0, 1]$. Unless $M$ has a vanishing Riemann tensor $R$, I would expect that there is no guarantee that $Z_s = s\tau_p^q(Y-X)$, where $\tau_p^q:T_pM\to T_qM$ denotes the parallel transport from $p$ to $q$ along the geodesic.
How can one estimate the difference $\tau_q^pZ_s - s(Y-X)\in T_pM$ in terms of $R$, or any other geometric objects?
Attempt By tiling the geodesic triangle with small rectangles of sides parallel to $X$ and $Y$ one could argue that there are contributions from each point from the image $\Sigma$ of the triangle under the exponential map. Hence, for a generic vector $Z$ (parallel transports are dropped for simplicity)
$$\Delta Z = \int_\Sigma R_p(X,Y)Z\ \text d\Sigma.$$