The propagation of an axisymmetric laser beam in a plasma column having a radially parabolic electron density distribution is examined. First, an extended paraxial procedure is developed for the case of an axially uniform waveguide. It is shown that the essential feature of an alternate focusing and defocusing beam is retained, but that the intensity distribution is cumulatively modified at the foci and at the outer portions of the beam as compared to that of the paraxial case. Second, some general features of paraxial beam propagation are examined for the case of axially varying waveguides. Finally, laser plasma coupling is examined for the case when laser heating generates a density distribution that is radially parabolic near the axis and when the energy absorbed over a focal length of a plasma lens is small. It is shown that stable or unstable beam propagation depends upon the relative magnitude of the density fluctuations which exist in the axial variation of the waveguides as a result of laser heating. When the fluctuations are small, the propagation is stable, and a simple algebraic expression is obtained which relates the beam diameter to the axially slow averaged variation in the waveguide. When the fluctuations are large, the propagation stability can be determined only by consistently combining plasma dynamics and beam propagation to interrelate the axial variation of the beam to that of the waveguide. In this case of beam propagation in a time‐varying waveguide, it is shown that the global stability of the propagation depends upon the initial fluctuation growth rate compared to the initial time rate of change in the radial curvature of the waveguide.