  In an earlier work [S. Kastha et al., PRD {\bf 98}, 124033 (2018)], we
developed the {\it parametrized multipolar gravitational wave phasing formula}
to test general relativity, for the non-spinning compact binaries in
quasi-circular orbit. In this paper, we extend the method and include the
important effect of spins in the inspiral dynamics. Furthermore, we consider
parametric scaling of PN coefficients of the conserved energy for the compact
binary, resulting in the parametrized phasing formula for non-precessing
spinning compact binaries in quasi-circular orbit. We also compute the
projected accuracies with which the second and third generation ground-based
gravitational wave detector networks as well as the planned space-based
detector LISA will be able to measure the multipole deformation parameters and
the binding energy parameters. Based on different source configurations, we
find that a network of third-generation detectors would have comparable ability
to that of LISA in constraining the conservative and dissipative dynamics of
the compact binary systems. This parametrized multipolar waveform would be
extremely useful not only in deriving the first upper limits on any deviations
of the multipole and the binding energy coefficients from general relativity
using the gravitational wave detections, but also for science case studies of
next generation gravitational wave detectors.
