For which boundaries of the Mandelbrot set are known to be parameterized by radical algebraic functions?

Question

For example, the boundary of period 1 can be parametrized by $X(1-X)$ for $$X=\frac{e^{\imath t}}{2},$$ period 2 can be parameterized by $X/2-1$, and period 3 boundaries are expressible as the solution of a cubic. Are there any other period boundaries that can be parameterized in radicals? They are all algebraic functions of $X$ though, that I am certain of!