**Abstract** : The numerical approximation of parametric partial differential equations D(u,y)=0 is a computational challenge when the dimension d of of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map u -> u(y) with an algebraic convergence rate that is independent of the parametric dimension d. The analysis used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for some classes parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z->u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying radii of the ellipses in each coordinate zj reflect the anisotropy of the solution map with respect to the corresponding parametric variables yj. This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case where d is infinite. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDE's that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector y in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on y, (iii) nonlinear, monotone parametric elliptic PDE's, and (iv) elliptic equations set on a domain that is parametrized by the vector y. We give general strategies that allows us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions.