  In this work we study the essential spectra of composition operators on
weighted Bergman spaces of analytic functions which might be termed as
"quasi-parabolic." This is the class of composition operators on
$A_{\alpha}^{2}$ with symbols whose conjugate with the Cayley transform on the
upper half-plane are of the form $\varphi(z)=$ $z+\psi(z)$, where $\psi\in$
$H^{\infty}(\mathbb{H})$ and $\Im(\psi(z)) > \epsilon > 0$. We especially
examine the case where $\psi$ is discontinuous at infinity. A new method is
devised to show that this type of composition operators fall in a C*-algebra of
Toeplitz operators and Fourier multipliers. This method enables us to provide
new examples of essentially normal composition operators and to calculate their
essential spectra.
