Parameter-induced stochastic resonance is an important method of detecting weak signal from noise, but under stable noise background, this method has not been reported. In this paper, we study the parameter-induced stochastic resonance in an overdamped system with stable noise. Our investigation discloses that the stochastic resonance can be realized by tuning the system parameter under stable noise background; when the nonlinear term parameter is turned, the resonant effect becomes weakened as the stability index decreases. But when the linear term parameter is turned, the resonant effect becomes strengthened as the stability index decreases. Our observation is significant for understanding the positive role of stable noise in weak signal detection, which is helpful for understanding the effects of different stable noises on stochastic resonance systems.