The Rusanov solver for solving hydrodynamic equations is one of the most robust schemes in the class of Riemann solvers. For special relativistic hydrodynamics, the robustness condition of the scheme is the most important property, especially for sufficiently high values of the Lorentz factor. At the same time, the Rusanov solver is known to be very dissipative. It is proposed to use a piecewise parabolic representation of physical variables to reduce the dissipation of the Rusanov scheme. Using this approach has made it possible to obtain a scheme with the same dissipative properties as Roe-type schemes and the family of Harten–Lax–van Leer schemes. Using the problem of the decay of a relativistic hydrodynamic discontinuity, it is shown that the present author’s version of the Rusanov scheme is advantageous in terms of reproducing a contact discontinuity. The scheme is verified on classical problems of discontinuity decay and on the problem of the interaction of two relativistic jets in the three-dimensional formulation.