Let $mathcal{K}$ be the space of properly embedded minimal tori in quotients of $R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $mathcal{K}$ is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in cite{ka4,ka6,mr3}. The degenerate limits of surfaces in $mathcal{K}$ are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples.