Sawtooth discharge is an important discharge mode in tokamak experiments. Based on ohmic discharge experiments with a parabolic current profile in the HL-2A tokamak and employing the MHD equilibrium code EFIT and stability code GATO, this study investigates the evolution of kink modes in toakamk sawtooth discharges by simulating and scanning the magnetohydrodynamic (MHD) equilibrium and stability at different axial safety factor (q₀) and poloidal beta (β_p). The results indicate that during the sawtooth quiescent phase, the q₀ of circular cross-section plasma typically ranges between 0.6 and 0.8, while the q₀ of divertor plasma is slightly less than 1, and it relaxes to slightly above 1 when the sawtooth wave collapses. When q₀<1, an m=1/n=1 internal kink mode appears in the plasma. If β_p>0.3, the mode becomes unstable. As β_p increases, the instability continues to develop, and at higher β_p, the growth of kink instability leads to internal disruptions, triggering sawtooth collapse. Sawtooth discharges are in a dynamic process where the plasma oscillates between unstable (when q₀<1) and stable (when q₀>1) states during each sawtooth cycle. As long as the operational β limit is not beyond, internal disruptions caused by kink instability do not lead to major disruptions. Whether the plasma disrupts or not depends on whether the kink mode continues to develop, so the operational β limit is determined on the base of the kink mode β limit when q₀≳1. Compared to non-sawtooth discharges with q₀≫1, a relatively higher maximum operational β can be achieved in sawtooth discharges. It was found that in discharge experiments with parabolic safety factor profiles, the operational β limit of sawtooth discharges was high compared to non-sawtooth discharges. We explained the internal disruptions and sawtooth collapses from the perspective of evolution of kink modes according to experimental and simulation results. The analysis highlighted the significance and application of sawtooth discharges in predicting the operational β limits of the device, demonstrating that sawtooth discharges are one of the ideal operational modes for modern tokamaks, including ITER.