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.