  We construct a family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$, the parabolic
quadric of $\textup{PG}(4,q)$, for $q\equiv 3\pmod 4$. The existence of
$\frac{(q-1)}{2}$-ovoids of $Q(4,q)$ was only known for $q=3, 7,$ or $11$. Our
construction provides the first infinite family of $\frac{(q-1)}{2}$-ovoids of
$Q(4,q)$.Along the way, we also give a construction of $\frac{q+1}{2}$-ovoids
in $Q(4,q)$ for $q\equiv 1\pmod 4$.
