We experimentally investigate drag modulation by finite-size particles in viscoelastic Taylor-Couette turbulence. Three spheroidal particle shapes, namely oblate, spherical and prolate particles, are considered. By combining global torque measurements with particle-distribution measurements, we identify two distinct particle-polymer-turbulence coupling regimes. At lower Reynolds number, the flow is particle-dominated: polymer elasticity only weakly modifies the carrier turbulence, while finite-size particles introduce additional particle-related stresses and locally enhance the particle-polymer interaction. In this regime, the particle-induced drag penalty increases monotonically with polymer concentration. At higher Reynolds number, the flow becomes polymer-dominated: stronger elasticity suppresses turbulent fluctuations and weakens the ability of particles to enhance angular momentum transport, leading to a non-monotonic dependence of the particle-induced drag penalty on polymer concentration. Particle shape further modulates this particle-polymer-turbulence coupling. Spherical particles generate larger torque than oblate and prolate particles, and this shape dependence is amplified in the viscoelastic fluid, originating from the fact that polymer stresses are sensitive to the local disturbance generated by particle geometry. Particle-distribution measurements provide the spatial signature of the two regimes: polymers have little influence at low Reynolds number, whereas at high Reynolds number they reduce the near-wall accumulation of spherical particles and make the axial distribution more strongly organized by Taylor vortices. These results reveal the regime-dependent coupling mechanisms in complex multiphase turbulence, where dispersed particles, polymer elasticity, turbulence, and secondary flows act as a coupled dynamical system to determine the particles and momentum transport.