A switchable and tunable dual-wavelength thulium-doped fiber laser (ST-DW-TDFL) based on the parallel filter is proposed and experimentally demonstrated in this study. In the proposed ST-DW-TDFL, there are two independent branches to form the parallel filter. One branch uses an all-fiber Sagnac interferometer embedded with a fiber Bragg grating (FBG-FSI) as the wavelength selection element, and the other branch uses a Fabry-Perot (FP) filter as the wavelength selection element. By adjusting two optical attenuators to change the light power of two branches, single and dual-wavelength output can be achieved and switched. On the basis of achieving dual-wavelength output, independent and continuous tuning of two wavelengths can be achieved by heating the FBG and adjusting the FP filter. The tuning range of one wavelength is up to 5.84 nm, while the tuning range of the other wavelength is up to 99.86 nm. The maximum wavelength variation and power fluctuation are less than 0.03 nm and 0.5 dB at room temperature. And the experimental results show that the proposed ST-DW-TDFL has flexible wavelength switching and tuning capabilities and good wavelength and power stabilities.