The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities of the quasi-parabolic structure equals one. This gives a new proof that the moduli space of vector bundles of coprime rank and degree is stably rational, a result originally due to Ballico, and the bound on the level is strong enough to deduce rationality in many cases, extending results of Newstead.