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In this paper we consider the C*-algebra$C^{*}(\{C_{\varphi}\}\cup\mathcal{T}(PQC(\mathbb{T})))/K(H^{2})$ generated byToeplitz operators with piece-wise quasi-continuous symbols and a compositionoperator induced by a parabolic linear fractional non-automorphism symbolmodulo compact operators on the Hilbert-Hardy space $H^{2}$. This C*-algebra iscommutative. We characterize its maximal ideal space. We apply our results tothe question of determining the essential spectra of linear combinations of aclass of composition operators and Toeplitz operators.

Parabolic Non-Automorphism Induced Toeplitz-Composition C*-Algebras with Piece-wise Quasi-continuous Symbols