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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 $\\mathcal{M}_{\\mathcal{A}}$. 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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 $\\mathcal{M}_{\\mathcal{A}}$. 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