In this paper we present new parallel versions of sequential Goertzel and Reinsch algorithms for calculating trigonometric sums. The new algorithms use a recently introduced, very efficient BLAS-based algorithm for solving linear recurrence systems with constant coefficients. To achieve their portability across different shared-memory parallel architectures, the algorithms have been implemented in Fortran 77 and OpenMP. We also present experimental results performed on a two processor Pentium III computer running under Linux operating system with Atlas as an efficient implementation of BLAS. The new algorithms are up to 60-90% faster than the equivalent sequential Goertzel and Reinsch algorithms, even on one processor.