Let \(\mathbb{C}\) denote the plane of complex numbers, and for real number \(v\), and analytic functions \(f\), \(g\) representable as \[ f(z)= \sum\{a_nz^n, n= 0,1,2,\dots\},\quad g(z)= \sum\{b_n z^n, n= 0,1,\dots,\}, \] let \((f|g)_v= \sum\{(a_n b_n(n+ 1)^{2v}, n= 0,1,\dots\}\), \(\|f\|^2_v= \sum\{|a_n|^2(n+ 1)^{2v}, n= 0,1,2,\dots\}\). Then the Hilbert space \(S_v\) is defined to be \(\{f\) analytic on \(\mathbb{C}:\|f\|_v< \infty\}\). If \(H\) is a Hilbert space with respect to an inner product and associated norm, and \(T\) is a bounded linear operator on \(H\), then \(T\) is said to be cyclic if, for some vector \(f\) in \(H\), the linear span of \(\{T^n(f), n= 0,1,2,\dots\}\) is a dense subspace of \(H\), \(T\) is said to be supercyclic if \(\{T^n(f), n=0,1,2,\dots\}\) is a dense subspace of \(H\), and \(T\) is said to be hypercyclic if \(\{\lambda T^n(f)\),\(\lambda\in\mathbb{C}\), \(n= 0,1,2,\dots\}\) is a dense subspace of \(H\). If \({\mathcal D}\) denotes the unit disk of the complex plane, if \(\varphi:{\mathcal D}\to{\mathcal D}\), and if the composition operator \({\mathcal C}_\varphi\) is defined by \({\mathcal C}_\varphi(f)(z)= f(\varphi(z))\), then it is shown in the main result of this paper that: if \(\varphi:{\mathcal D}\to{\mathcal D}\) is a parabolic non-automorphism, then \({\mathcal C}_\varphi\) is not supercyclic in any of the weighted Dirichlet spaces \(S_v\). The case \(v=0\) of the result is indicated as contained in a separate paper to be published. In particular, it is stated in the introduction of this paper that the result completes statemens relating to the supercyclic behaviour of linear `fractional' composition operators in Hardy space \({\mathbf H}^2\).