They can almost be parameterized by Abelian differentials: Take a flat $SU(n)$-connection and its monodromy along two generators of the first fundamental group of the torus. These are two commuting $SU(n)$ matrices and therefore they diagonalize simultaneously. Their common eigenlines define parallel line subbundles (take the parallel transport of those eigenlines), hence every flat $SU(n)$-connection is gauge equivalent to the direct sum of flat unitary line bundle connections on that torus. (Their product is the trivial connection on the determinant bundle.) Every unitary flat line bundle connection on the torus $\mathbb C/\Gamma$ is gauge equivalent to some \begin{equation} (1)\;\;\;\;\;\nabla=d+\alpha dz-\bar\alpha d\bar z\end{equation} for some $\alpha.$ Hence, your flat $SU(n)$ connection is determinend by $n$ abelian differentials $$\alpha_1 dz,..,\alpha_n dz$$ satisfying the additional determinant condition: $$\sum \alpha_i=0.$$
As the parallel line bundles have no natural ordering, you should factor out the Weyl group. Additionally, there is a lattice $\tilde\Gamma\subset\mathbb C$ with the property that two connections of the form (1) for $\alpha_1,\alpha_2\in\mathbb C$ are gauge equivalent if and only if their difference is in $\tilde\Gamma.$
It should be noted that most people (as the authors of [1]) seem to prefer to parametrize the flat unitary line bundles in terms of the underlying holomorphic structure, hence by the jacobian of the torus.
