Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ admitting a 3-dimensional group

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.