Based on the analysis of the muiltiplicative structure of Koszul algebras given by Buchweitz et al, a sufficient and necessary condition for the multiplicative structure of Hochschild cohomology rings of Koszul algebras to be essentially the juxtaposition of parallel paths is obtained. As an application, it is used to give an alternative proof of Bustamante's results that the multiplicative structures of Hochschild cohomology rings of quadratic triangular string algebras are trivial, which improves the original proof.