 Solution

The parabolic basin $A$ is open and lies in $F(g)$. Suppose $z\in \partial A\cap F(g)$ and let $U$ be the Fatou component containing $z$. A small neighborhood of $z$ in $U$ meets $A$. On one point of $U$, the iterates eventually enter the chosen attracting petal and converge to zero along its attracting vector. Normality and the identity theorem for the limiting iterates make the same true throughout the connected component $U$. Hence $U\subseteq A$, which is impossible for a boundary point. Therefore the parabolic basin satisfies