Owing to their excellent performance and large design space, curvilinear fiber-reinforced composite structures have gained considerable attention in engineering fields such as aerospace and automobile. In addition to the stiffness and strength of such structures, their stability also needs to be taken into account in the design. This study proposes a level-set-based optimization framework for maximizing the buckling load of curvilinear fiber-reinforced composite structures. In the proposed method, the contours of the level set function are used to represent fiber paths. For a composite laminate with a certain number of layers, one level set function is defined by radial basis functions and expansion coefficients for each layer. Furthermore, the fiber angle at an arbitrary point is the tangent orientation of the contour through this point. In the finite element of buckling, the stiffness and geometry matrices of an element are related to the fiber angle at the element centroid. This study considers the parallelism constraint for fiber paths. With the sensitivity calculation of the objective and constraint functions, the method of moving asymptotes is utilized to iteratively update all the expansion coefficients regarded as design variables. Two numerical examples under different boundary conditions are given to validate the proposed approach. Results show that the optimized curved fiber paths tend to be parallel and equidistant regardless of whether the composite laminates contain holes or not. Meanwhile, the buckling resistance of the final design is significantly improved.