This question is concerned with Section 2.5. of Amari and Nagaoka's Information geometry book. Let me give some background first. Let $\mathcal{P}$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in \mathcal{P}$ is thought of a point in $\mathbb{R}^{\mathcal{X}}:=\{A \big|~A:\mathcal{X}\to \mathbb{R}\}$. Hence the tangent space $T_p(\mathcal{P})$ at each point $p$ is $\{A\big|\sum_x A(x)=0\}$. For two tangent vectors $X,Y\in T_p(\mathcal{P})$, a Riemannian metric (the Fisher information metric) is given by $$\langle X,Y\rangle^{(e)}_p:=E_p[X^{(e)}Y^{(e)}],$$ where $X^{(e)}(x)=X(x)/p(x)$ is called the exponential representation of $X$ at $p$. One can show that $\{X^{(e)}\big| X\in T_p(\mathcal{P})\}=\{A\big|E_p[A]=0\}$.
Now, let $[\xi^i]$ be a coordinate system on $\mathcal{P}$ with $\{\partial_i:=\partial/\partial_{\xi_i}\}$ being the natural basis and $(\partial_i^{(e)})_{p_{\xi}}:=\partial_i\log p_{\xi}$. The exponential connection ($e$-connection) on $\mathcal{P}$ with respect to this coordinate system is given by \begin{eqnarray} \Gamma_{ij,k} &:=& \langle \nabla_{\partial_i}^{(e)}\partial_j,\partial_k\rangle\\ &=& \sum_x\partial_i(\partial_j\log p_{\xi}(x))\cdot \partial_k p_{\xi}(x)\\ &=& E_{p_{\xi}}[\partial_i(\partial_j\log p_{\xi})~\partial_k \log p_{\xi}]. \end{eqnarray}
They claim the following regarding the $e$-connection:
\begin{eqnarray} \label{eq1} \tag{1} \Pi_{p,q}^{(e)}(X)=X'\Longleftrightarrow X_q'^{(e)}=X_p^{(e)}-E_q[X_p^{(e)}], \end{eqnarray}
i.e., the tangent vector $X$ at $p$ is parallelly translated to $X'$ at $q$ iff $X_q'^{(e)}=X_p^{(e)}-E_q[X_p^{(e)}]$.
The 'if' part seems clear. Indeed, if $X:p\mapsto X_p$ is an arbitrary vector field on $\mathcal{P}$ such that $X_q^{(e)}=X_p^{(e)}-E_q[X_p^{(e)}]$, then $X_q^{(e)}=F-E_q[F]$ for all $q\in \mathcal{P}$, where $F$ is some real valued function on $\mathcal{X}$. hence $\partial_i(X_p^{(e)}(x)) = -\partial_i(\mathbb{E}_p[F])$, a constant as a function of $x$, for any $p$, and hence \begin{eqnarray} \langle \nabla_{\partial_i}^{(e)}X, \partial_k\rangle_p &=&\mathbb{E}_p[ \partial_i(X_p^{(e)})(\partial_k)_p^{(e)}]\\ & = & -\partial_i(\mathbb{E}_p[F])\cdot \mathbb{E}_p[(\partial_k)_p^{(e)}]\\ & = & 0, \end{eqnarray} hence $X$ is $e$-parallel.
Although they seems to have concluded the entire claim by the above argument, I'm not convinced. My question is, does the 'only if' part of (\ref{eq1}) also follow from this? Or is it easy to see from someother argument? Any help is greatly appreciated.