We present 1.4 pc resolution observations of 256 GHz nuclear radio continuum and HCN ($J=3\to 2$) in the molecular torus of NGC 1068. The integrated radio continuum emission has a flat spectrum consistent with free-free emission and resolves into an X-shaped structure resembling an edge-brightened bicone. HCN is detected in absorption against the continuum, and the absorption spectrum shows a pronounced blue wing that suggests a high-velocity molecular outflow with speeds reaching 450 km/s. Analysis of the off-nucleus emission line kinematics and morphology reveals two nested, rotating disk components. The inner disk, inside $r\sim 1.2$ pc, has kinematics consistent with the nearly edge-on, geometrically thin water megamaser disk in Keplerian rotation around a central mass of $1.66\times 107M\odot$. The outer disk, which extends to $\sim 7$~pc radius, counter-rotates relative to the inner disk. The rotation curve of the outer disk is consistent with rotation around the same central mass as the megamaser disk but in the opposite sense. The morphology of the molecular gas is asymmetric around the nuclear continuum source. We speculate that the outer disk formed from more recently introduced molecular gas falling out of the host galaxy or from a captured dwarf satellite galaxy. In NGC 1068, we find direct evidence that the molecular torus consists of counter-rotating and misaligned disks on parsec scales.