Prefixes of minimal factorisations of a cycle

We give a bijective proof of the fact that the number of k-prefixes of minimal factorisations of the n-cycle (1...n) as a product of n-1 transpositions is n^{k-1}\binom{n}{k+1}. Rather than a bijection, we construct a surjection with fibres of constant size. This surjection is inspired by a bijection exhibited by Stanley between minimal factorisations of an n-cycle and parking functions, and by a counting argument for parking functions due to Pollak.