Update: I have made some progress by following Alexandre Eremenko's suggestion to examine the orbits of the critical points. I would like to share an example application of this technique for identifying (rational) neutral cycles.
Consider the family of functions $f:z\to z - p(z)/p'(z)$, describing Newton-Raphson iteration on $p$, where $p$ is a cubic polynomial with simple roots. An element $f$ of this family has three attracting fixed points in $\mathbb{C}$, the roots of $p$, and is conjugate by a Möbius transformation to a function with an attracting fixed point at -1, an attracting fixed point at 1, and an attracting fixed point at $\lambda\in\mathbb{C}$. The following image is a visualization of types of cycles occurring in $f_\lambda$, for $\lambda$ in the axis aligned square of side length 0.04 with top left corner $-0.834 + 0.861 i$. The only candidate for a critical point of $f_\lambda$ is $\alpha = \lambda/3$.
In the image a positive green color component at a point $\lambda$ indicates that $f_\lambda^n(\alpha)$ converges to one of the fixed points (1, -1, or $\lambda$). Brighter green indicates slower convergence. The green regions are typical of Newton basin renderings. Points $\lambda$ having no green color component, but positive red and blue color components, indicate that $f_\lambda^n(\alpha)$ converges to a cycle of period greater than 1. In this case the cycle may be neutral or attracting. Brighter purple indicates slower convergence, so the bright purple regions indicate values of $\lambda$ such that $f_\lambda$ probably has neutral periodic points and $F(f_\lambda)$ might have parabolic components or Siegel disk components.
