fractals - Hausdorff dimension of higher powers of the Mandebrot set ?

Yes, it does. See the full statement of Theorem 2 on page 6. The assumptions of the theorem are:

Suppose that a rational map $f_0$ of degree $d (> 1)$ has a parabolic fixed
point ζ with multiplier exp(2πip/q) ($p, q inmathbb{Z}, mathit{gcd}(p, q) = 1$) and that the immediate parabolic basin of ζ contains only one critical point of $f_0$.

This is the case for $z^d+c$.