  Given a field $k$ and a finite group $H$, {\it{an $H$-parametric extension
over $k$}} is a finite Galois extension of $k(T)$ of Galois group containing
$H$ which is regular over $k$ and has all the Galois extensions of $k$ of group
$H$ among its specializations. We are mainly interested in producing non
$H$-parametric extensions, which relates to classical questions in inverse
Galois theory like the Beckmann-Black problem and the existence of one
parameter generic polynomials. We develop a general approach started in a
preceding paper and provide new non parametricity criteria and new examples.
