Let M be a real hypersurface of a complex space form with almost contact metric structure $({phi},{xi},{eta},g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_{xi}=R({cdot},{xi}){xi}$ is ${xi}$-parallel. In particular, we prove that the condition ${ abla}_{xi}R_{xi}=0$ characterize the homogeneous real hypersurfaces of type A in a complex: projective space $P_n{mathbb{C}}$ or a complex hyperbolic space $H_n{mathbb{C}}$ when $g({ abla}_{xi}{xi},{ abla}_{xi}{xi})$ is constant and not equal to -c/24 on M, where c is a constant holomorphic sectional curvature of a complex space form.