For the thrice-punctured sphere, there is a generating set where both generators are parabolic.  For the once-punctured only the commutator (and its conjugates) is parabolic.  Hence any element that can be extended to give a generating is hyperbolic. 

Thinking in the upper half-plane model of $\mathbb H$, the Farey set is the rational points of the real line (plus the point at infinity).  If $X$ is a punctured hyperbolic surface then the lifts of the ideal points of $X$ likewise form a dense set in the real line.  If $X$ is a thrice-punctured sphere then, after possibly conjugating the deck group by an isometry of $\mathbb H$ the lifts of the punctures are the Farey set.  However, for any other surface $X$ this only happens if the _modulus_ of $X$ is carefully chosen.