  In this paper we focus on the description of the automorphism group
$\Gamma_{\parallel}$ of a Clifford-like parallelism $\parallel$ on a
$3$-dimensional projective double space
$\bigl(\mathbb{P}(H_F),{\mathrel{\parallel_{\ell}}},{\mathrel{\parallel_{r}}}\bigr)$
over a quaternion skew field $H$ (with centre a field $F$ of any
characteristic). We compare $\Gamma_{\parallel}$ with the automorphism group
$\Gamma_{\ell}$ of the left parallelism $\mathrel{\parallel_{\ell}}$, which is
strictly related to $\mathrm{Aut}(H)$. We build up and discuss several examples
showing that over certain quaternion skew fields it is possible to choose
$\parallel$ in such a way that $\Gamma_{\parallel}$ is either properly
contained in $\Gamma_{\ell}$ or coincides with $\Gamma_{\ell}$ even though
${\parallel} \neq{\mathrel{\parallel_{\ell}}}$.
