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