In this paper, we study the nonlinear optical bistability (OB) in a symmetrical multilayer structure. This multilayer structure is constructed by embedding a nonlinear three-dimensional Dirac semimetal (3D DSM) into a Fabry–Perot cavity composed of one-dimensional photonic crystals. The OB phenomenon stems from the third order nonlinear conductivity of 3D DSM. The local field of resonance mode could enhance the nonlinearity and reduce the thresholds of OB. This structure achieves the tunability of OB due to the fact that the transmittance could be modulated by the Fermi energy. It is found that the OB threshold and threshold width could be remarkably reduced by increasing the Fermi energy of the 3D DSM. Besides, we also found that the OB curve depends heavily on the angle of incidence of the incident light, the structural parameters of the Fabry–Perot cavity, and the position of the 3D DSM inside the cavity. After parameter optimization, we obtained OB with a threshold of ${10}^6 \mathrm{V}/\mathrm{m}$. We believe this simple multilayer structure could provide a reference idea for realizing low-threshold and tunable all-optical switching devices.