This is a joint work with Cui Guizhen. A parabolic point is a periodic point with multiplicity m that is at least 2. A perturbation will break the point into a certain number of points with total multiplicity m. A map f with parabolic points is often a common boundary parameter point of several distinct hyperbolic components. Some of them have a dynamics similar to that of f. We consider them as 'dynamically stable' perturbations of f. We will construct converging parameter rays within each stable perturbation. For this we will not use the usual approach of analysing the parametrization, instead we will use surgery to construct the path of maps with desired dynamical properties.