  We generalize a result of F.\ Legrand about the existence of non-parametric
Galois extensions for a given group $G$. More precisely, for a $K$-regular
Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an
extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a
root of $g(X)-t$ for some rational function $g\in K(X)$). We then show that if
$F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field
$K$, then for any degree $k\ge 2$ and almost all (in a density sense) rational
functions $g$ of degree $k$, the translate of $F$ by a root field of $g(X)-t$
over $K(t)$ is non-$G$-parametric, i.e.\ not all Galois extensions of $K$ with
group $G$ arise as specializations of $F(s)|K(s)$.
