  Let $L/K$ be a finite separable extension of fields of degree $n$, and let
$E/K$ be its Galois closure. Greither and Pareigis showed how to find all
Hopf-Galois structures on $L/K$. We will call a subextension $L'/K$ of $E/K$
\textit{parallel} to $L/K$ if $[L':K]=n$.
  In this paper, we investigate the relationship between the Hopf-Galois
structures on an extension $L/K$ and those on its parallel extensions. We give
an example of a transitive subgroup corresponding to an extension admitting a
Hopf-Galois structure but that has a parallel extension admitting no
Hopf-Galois structures. We show that once one has such a situation, it can be
extended into an infinite family of transitive subgroups admitting this
phenomenon. We also investigate this fully in the case of extensions of degree
$pq$ with $p,q$ distinct odd primes, and show that there is no example of such
an extension admitting the phenomenon.
