  Over any field $\mathbb K$, there is a bijection between regular spreads of
the projective space ${\rm PG}(3,{\mathbb K})$ and $0$-secant lines of the
Klein quadric in ${\rm PG}(5,{\mathbb K})$. Under this bijection, regular
parallelisms of ${\rm PG}(3,{\mathbb K})$ correspond to hyperflock determining
line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is
defined to be \emph{pencilled} if it is composed of pencils of lines. We
present a construction of pencilled hfd line sets, which is then shown to
determine all such sets. Based on these results, we describe the corresponding
regular parallelisms. These are also termed as being \emph{pencilled}. Any
Clifford parallelism is regular and pencilled. From this, we derive necessary
and sufficient algebraic conditions for the existence of pencilled hfd line
sets.
