  Betten and Riesinger constructed Parallelisms of $\mathop{\rm PG}(3,\mathbb
R)$ with automorphism group $\mathop{\rm SO}(3,\mathbb R)$ by applying the
reducible $\mathop{\rm SO}(3,\mathbb R)$-action to a rotational Betten spread.
This was generalized by the present author so as to include oriented
parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger
class of examples was produced. Here we show that, apart from Clifford
parallelism, these are the only topological parallelisms admitting an
automorphism group of dimension 3 or larger. In particular, we show that a
topological parallelism admitting the irreducible action of $\mathop{\rm
SO}(3,\mathbb R)$ is Clifford.
