We consider possible signatures for Yukawa gravity within theGalactic Central Parsec, based on our analysis of the S2 star orbitalprecession around the massive compact dark object at the GalacticCentre, and on the comparisons between the simulated orbits in Yukawa gravityand two independent sets of observations. Our simulations resulted in strongconstraints on the range of Yukawa interaction Λ and showed that itsmost probable value in the case of S2 star is around 5000 - 7000 AU. Atthe same time, we were not able to obtain reliable constrains onthe universal constant δ of Yukawa gravity, because thecurrent observations of S2 star indicated that it may be highlycorrelated with parameter Λ in the range (0 < δ < 1).For δ > 2 they are not correlated. However, the same universal constant whichwas successfully applied to clusters of galaxies and rotation curvesof spiral galaxies (δ = 1/3) also gives a satisfactory agreement withthe observed orbital precession of the S2 star, and in that casethe most probable value for the scale parameter is Λ ≈ 3000±1500 AU. Also, the Yukawa gravity potential induces precession of S2 starorbit in the same direction as General Relativity for δ > 0 and forδ < −1, and in the opposite direction for −1 < δ < 0. The futureobservations with advanced facilities, such as GRAVITY or/and European ExtremelyLarge Telescope, are needed in order to verify these claims.