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Leveraging a high-resolution 3D dust map of the solar neighborhood from Edenhofer et al. (2024), we derive a new 3D model for the dust-traced surface of the Local Bubble, the supernova-driven cavity surrounding the Sun. We find that the surface of the Local Bubble is highly irregular in shape, with its peak extinction surface falling at an average distance of 170 pc from the Sun (spanning 70-600+ pc) with a typical thickness of 35 pc and a total dust-traced mass of $(6.0 \pm 0.7) \times 10^5 \ \rm{M}_{\odot}$. The Local Bubble displays an extension in the Galactic Northern hemisphere that is morphologically consistent with representing a "Local Chimney." We argue this chimney was likely created by the "bursting" of this supernova-driven superbubble, leading to the funneling of interstellar medium ejecta into the lower Galactic halo. We find that many well-known dust features and molecular clouds fall on the surface of the Local Bubble and that several tunnels to other adjacent cavities in the interstellar medium may be present. Our new, parsec-resolution view of the Local Bubble may be used to inform future analysis of the evolution of nearby gas and young stars, the investigation of direct links between the solar neighborhood and the Milky Way's lower halo, and numerous other applications.