This research conducts a systematic examination of the nonlinear dynamic behaviour exhibited by curved single-walled carbon nanotubes (CSWCNTs) under parametric excitation while embedded in a nonlinear viscoelastic medium. The investigation employs nonlocal Euler–Bernoulli beam theory to formulate a size-dependent continuum model that incorporates nonlocal elasticity effects. The derived nonlinear partial differential governing equation undergoes reduction through single-mode Galerkin decomposition, transforming the system into a solvable set of ordinary differential equations. Applying the method of multiple scales, we analytically derive closed-form solutions characterising the primary resonance response. A comprehensive parametric analysis evaluates the governing influence of multiple key parameters: (i) self-sustaining excitation coefficients, (ii) geometric curvature amplitudes, (iii) parametric forcing magnitude, (iv) thermal gradients, (v) magnetic field intensities, and (vi) frequency detuning parameters. The study elucidates fundamental mechanisms governing stability boundaries in CSWCNTs, explicitly accounting for quadratic nonlinearities induced by curvature effects, geometric nonlinearities manifesting as cubic displacement terms, and velocity-dependent nonlinearities contributed by viscoelastic substrate. These findings establish critical insights into the interplay between multiphysics excitations and nonlinear response characteristics, advancing predictive capabilities for nanotube-based nanoelectromechanical systems operating under complex dynamic loading conditions.