Let $f \in \mathbb{Z}[X,Y,Z]$ be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by $f=0$ has a~function field isomorphic to the rational function field $\mathbb{Q} (T)$. We show that all integral solutions of the Diophantine equation $f=0$ (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a~single triple of polynomials with integer coefficients.