For $q_c = z^2+c$ with $c\in \mathcal{M}$, the Mandelbrot set, if $c_1$ and $c_2$ are in the same bulb, will the Julia sets have the same lamination. So, for example, will all Douady rabbits have the same lamination regardless of the precise value of $c$?
Edit: I believe this is true but I'm not confident enough in my knowledge to say definitely.
Yes, it is true. Inside a connected component of the interior of the Mandelbrot set, all quadratic polynomials $q_c$ are topologically conjugate on their Julia set (which in particular implies that they are homeomorphic). The laminations are defined in terms of the landing of external rays; the conjugacies extend past the Julia set to the closure of the basin of infinity, and maps external rays to external rays of the same angle. So the lamination is the same.