Property of parallel transport

Question

Suppose I have three points $p_0,p_1,p_2$ on a regular surface $M$ in $\mathbb{R}^3$ and all the objects we use in the following are well defined. Denote by $PT$ the parallel transport from $T_{p_0}M$ to $T_{p_1}M$ along the geodesic connecting $p_0$ and $p_1$. Then, I know that $$\log_{p_1}p_0 = - PT(\log_{p_0}p_1).$$ Does the equation $$\log_{p_1}p_2 = PT(\log_{p_0}p_2)-PT(\log_{p_0}p_1) \in T_{p_1}M$$ also hold? If this is not the case, is there any other way to express $\log_{p_1}p_2$ by $\log_{p_0}p_2$ and $\log_{p_0}p_1$?