  In this paper, for fibrations $p:E\to B$, $p':E'\to B$ over $B$ and a free
involution $\tau:E\to E$ which satisfy the equality $p\circ\tau=p$, we discover
a connection between the sectional category of the double covers $q:E\to
E/\tau$ and $q^Y:F_{p'}(E',2)\to F_{p'}(E',2)/\mathbb{Z}_2$ from the
$2$-ordered fibre-wise configuration space $F_{p'}(E',2)$ to its unordered
quotient $ F_{p'}(E',2)/\mathbb{Z}_2$, and the parametrized Borsuk-Ulam
property (PBUP) for the triple $\left((p,\tau);p'\right)$. Explicitly, we
demonstrate that the triple $\left((p,\tau);p'\right)$ satisfies the PBUP if
the sectional category of $q$ is bigger than the sectional category of
$q^{p'}$. This property connects a standard problem in parametrized Borsuk-Ulam
theory to current research trends in sectional category. As applictions of our
results, we explore the PBUP for $E, E'$ one of the following fibrations:
trivial fibration, Hopf fibration and the Fadell-Neuwirth fibration.
