The authors discuss fast algorithms for the evolution of singular integral operators, in particular the singular integral transform \[ T_m h(\sigma)=-{1\over \pi} \iint_{B(0;1)} {h(\zeta)\over(\zeta- \sigma)^m} d\xi d\eta,\qquad \zeta= \xi+ i\eta \] which arises in solving Beltrami’s equation. Here \(h\) is a complex valued function of \(\sigma\) defined in the open unit disk \(B(0;1)\). These fast algorithms are based on a fast Fourier transform and some recursive relations. The authors develop a parallel version for these fast algorithms by reformulating in a novel way the inherently sequential recurrences. The resulting method uses only a linear neighbor to neighbor communication path is therefore suitable to many distributed memory architectures. The algorithms are worked out in great detail, including data distribution issues, complexity computations and numerical tests.