Topological photonics has emerged as a route to robust optical circuitry protected against disorder(1,2) and now includes demonstrations such as topologically protected lasing(3-5) and single-photon transport(6). Recently, nonlinear optical topological structures have attracted special theoretical interest(7-11), as they enable tuning of topological properties by a change in the light intensity(7,12) and can break optical reciprocity(13-15) to realize full topological protection. However, so far, non-reciprocal topological states have only been realized using magneto-optical materials and macroscopic set-ups with external magnets(4,16), which is not feasible for nanoscale integration. Here we report the observation of a third-harmonic signal from a topologically non-trivial zigzag array of dielectric nanoparticles and the demonstration of strong enhancement of the nonlinear photon generation at the edge states of the array. The signal enhancement is due to the interaction between the Mie resonances of silicon nanoparticles and the topological localization of the electric field at the edges. The system is also robust against various perturbations and structural defects. Moreover, we show that the interplay between topology, bianisotropy and nonlinearity makes parametric photon generation tunable and non-reciprocal. Our study brings nonlinear topological photonics concepts to the realm of nanoscience.