This research article holds two main purposes. The first purpose is obtaining the optical soliton solutions of the (1 + 1) -dimensional dispersive Schrödinger–Hirota equation in the presence of spatio-temporal dispersion with parabolic law nonlinearity. The second is scrutinizing the impact of the parameters. We utilized the new Kudryashov scheme which is the recent effective, efficient, and easily applicable technique. Performing the wave transformation the nonlinear ordinary differential form (NLODF) is gained. Then utilizing the new Kudryashov method the polynomial structure and its solution sets are derived. Selecting the appropriate solution set, establishing the solution function, providing the main equation, linear modulation stability analysis, and graphical presentations in 3D, and 2D are the issues performed in the remaining parts of the article.