  Given fields $k \subseteq L$, our results concern one parameter
$L$-parametric polynomials over $k$, and their relation to generic polynomials.
The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize
all Galois extensions of $L$ of group $G$ via specialization of $T$ in $L$, and
the latter are those which are $L$-parametric for every field $L \supseteq k$.
We show, for example, that being $L$-parametric with $L$ taken to be the single
field $\mathbb{C}((V))(U)$ is in fact sufficient for a polynomial $P(T, Y) \in
\mathbb{C}[T][Y]$ to be generic. As a corollary, we obtain a complete list of
one parameter generic polynomials over a given field of characteristic 0,
complementing the classical literature on the topic. Our approach also applies
to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer
conjecture, we provide one parameter families of affine curves over number
fields, all with a rational point, but with no rational generic point.
