Let 𝒦 be the space of properly embedded minimal tori in quotients of ℝ3 by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that 𝒦 is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in [9, 10, 15]. The degenerate limits of surfaces in 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.