The stationary-state theory of solar and stellar winds of E. N. Parker is examined for its dynamical stability properties and for the response of its critical transonic flow-curve to arbitrary first-order perturbations. The ideal gas law and the hydrodynamical equations of motion and continuity are employed, while the temperature is phenomenologically related to the density by a polytrope law. A set of linearized coupled differential equations is obtained for the independent spatial and temporal modes. By choosing boundary conditions such that disturbances vanish at the source of the flow, it is shown that there are no non-trivial solutions of the perturbation equations. This yields the important conclusion that Parker's steady-state theory is dynamically stable. Stationary-state disturbances are examined with spherical symmetry as asymmetry. With symmetry, a complete specification of first-order variables in terms of zero-order variables is obtained. For asymmetric disturbances, it is shown that special conditions must be satisfied by the variables at the base of the flow; exact solutions are given in the limit of asymptotically large radial distance. The transonic point of the zero-order flow is a singular point of the perturbation equations. This leads to the interpretation that the zero-order flow is unstable under all time-dependent boundary perturbations which do not meet the (specified) regularity condition at its transonic point.