This paper deals with the existence of bursting, mixed-mode oscillations (MMOs) and horseshoe chaos in a mixed Rayleigh–Liénard oscillator with asymmetric double well potential driven by parametric periodic damping and external excitations. The dynamical behaviours of the considered model, when the exciting frequency is much smaller than the natural frequency, are studied using bifurcation diagrams, Lyapunov exponent diagrams,time histories and phase portraits. It is found that our model displays various bursting and mixed-mode oscillations. Moreover, various mixed-mode oscillation routes to chaos occur in the system. It is also found that the system exhibits two or three coexisting behaviours of attractors when the parametric damping coefficient evolves. On the other hand, the analytical criterion for the appearance of horseshoes chaos is derived using the Melnikov method.A convenient demonstration of the accuracy of the method is obtained from the fractal basin boundary. It is noted that the increase of the impure quadratic damping coefficient, cubic damping coefficient, asymmetric term and the amplitude of the external excitation accentuates the chaotic behaviour of the system. However, the behaviour of the system becomes regular as the parametric damping coefficient increases.