  Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight
vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq
1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank
two parabolic vector bundles with trivial determinant on $\big(\mathbb{P}^1,
p_1,\dots , p_n\big)$ which are semistable with respect to $\mathcal{A}$. Under
some conditions on the weights, we determine and give a modular interpretation
for the automorphism group of the moduli space $\mathcal{M}_{\mathcal{A}}$. It
is isomorphic to $\left(\frac{\mathbb{Z}}{2\mathbb{Z}}\right)^{k}$ for some
$k\in \{0,\dots, n-1\}$, and is generated by admissible elementary
transformations of parabolic vector bundles. The largest of these automorphism
groups, with $k=n-1$, occurs for the central weight $\mathcal{A}_{F}=
\left(\frac{1}{2},\dots,\frac{1}{2}\right)$. The corresponding moduli space
${\mathcal M}_{\mathcal{A}_F}$ is a Fano variety of dimension $n-3$, which is
smooth if $n$ is odd, and has isolated singularities if $n$ is even.
