  Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a
$G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with
Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$
with Galois group $G$ among its specializations. We are mainly interested in
producing non-$G$-parametric extensions, which relates to classical questions
in inverse Galois theory like the Beckmann-Black problem. Building on a
strategy developed in previous papers, we show that there exists at least one
non-$G$-parametric extension over $F$ for a given non-trivial finite group $G$
and a given number field $F$ under the sole necessary condition that $G$ occurs
as the Galois group of a Galois extension $E/F(T)$ with $E/F$ regular.
