  Given two parallelisms of a projective space we describe a construction,
called blending, that yields a (possibly new) parallelism of this space. For a
projective double space $(\mathbb{P},\parallel_\ell,\parallel_r)$ over a
quaternion skew field we characterise the "Clifford-like" parallelisms, i.e.
the blends of the Clifford parallelisms $\parallel_\ell$ and $\parallel_r$, in
a geometric and an algebraic way. Finally, we establish necessary and
sufficient conditions for the existence of Clifford-like parallelisms that are
not Clifford.
