The free discrete subgroups consist of the closure of the Riley slice of Schottky space (notice that one may assume $Re(\alpha)\geq 0, Im(\alpha)\geq 0$, since $\alpha \mapsto -\alpha, \overline{\alpha}$ preserves discreteness).
Here's a picture of the first quadrant of the Riley slice:
The exterior of the black fractal represents free discrete groups that are generalized Schottky groups. These have been investigated in detail by Akiyoshi, Sakuma, Wada, and Yamashita. The black curve consists of geometrically finite groups with a cusp or degenerate groups. By the density conjecture, it is known that all free two-parabolic generator groups lie in the boundary of the Riley slice. Moreover, it is known that there is a group realizing each ending lamination parameterized by $\mathbb{R/Z}-\mathbb{Q/Z}$.
In the exterior of the Riley slice, there are many more non-free discrete groups, such as those corresponding to 2-bridge links. In fact, all of the torsion-free discrete groups correspond to 2-bridge links. Note that for 2-bridge links, although they are known to be generated by 2 parabolics, the precise value of $\alpha$ which gives the parabolics has not been determined. One may see an example of this for the twist knots, for which $\alpha$ has been worked out by Hoste and Shanahan.
For the discrete groups with torsion, see also some incomplete notes of mine.