We study the dissipative bistable Duffing oscillator with equal energy wells and observe fractal patterns in the parameter space of driving frequency, forcing amplitude, and damping ratio. Our numerical investigation reveals the Hausdorff fractal dimension of the boundaries that separate the oscillator's intrawell and interwell behaviors. Furthermore, we categorize the interwell behaviors as three steady-state types: switching, reverting, and vacillating. While fractal patterns in the phase space are well known and heavily studied, our results point to another research direction about fractal patterns in the parameter space. Another implication of this study is that the vibration of a continuous bistable system modeled using a single-mode approximation also manifests fractal patterns in the parameter space. In addition, our findings can guide the design of next-generation bistable and multistable mechanical metamaterials.