A paired many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. A two-dimensional $m{ imes}n$ torus is a graph defined as a product of two cycles $C_m$ and $C_n$ of length m and n, respectively. In this paper, we show that an $m{ imes}n$ torus with $m{geq}3$ and odd $n{geq}3$ has a 2-DPC joining any two source-sink pairs of vertices. This result is optimal in a sense that an $m{ imes}n$ torus does not always have a 3-DPC and that the graph with one faulty vertex or edge does not always have a 2-DPC.