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 ( q_0 ) and poloidal beta ( \beta _\textp ). The results indicate that during the sawtooth quiescent phase, the q_0 of circular cross-section plasma typically ranges between 0.6 and 0.8 , while the q_0 of divertor plasma is slightly less than 1, and it relaxes to slightly above 1 when the sawtooth wave collapses. When q_0\lt 1 , an m=1/n=1 internal kink mode appears in the plasma. If \beta _\textp> 0.3 , the mode becomes unstable. As \beta _p increases, the instability continues to develop, and at higher \beta _\textp , 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 q_0\lt 1 ) and stable (when q_0> 1 ) states during each sawtooth cycle. As long as the operational \beta 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 \beta limit is determined on the base of the kink mode \beta limit when q_0\gtrsim 1 . Compared to non-sawtooth discharges with q_0\gg 1 , a relatively higher maximum operational \beta can be achieved in sawtooth discharges. It was found that in discharge experiments with parabolic safety factor profiles, the operational \beta 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 \beta limits of the device, demonstrating that sawtooth discharges are one of the ideal operational modes for modern tokamaks, including ITER.