Recent high-resolution mapping of the circumnuclear regions of active galactic nuclei (AGNs) has revealed the existence of geometrically thin nuclear disks, in general randomly oriented with respect to their galaxy hosts. These molecular tori have typical radii of 10 pc, and contain a few 107 M⊙ of H2, with H2 column densities between 1023 and 1025 cm−2. We mapped two of the most massive of these molecular tori with a higher resolution, in order to unveil their morphology and kinematics, their possible warp and clumpiness, and derive their stability and lifetime. We used the highest resolution possible with ALMA (16 km baseline) in Band 7, taking into account for mapping CO(3─2) and HCO+(4─3) the compromise between sensitivity and resolution. New features are discovered at the high resolution, obtained with a beam of 0.015″, equivalent to ∼1 pc scale, at their ∼15 Mpc distance. The molecular torus in NGC 613 appears similarly to a ring, depleted in molecular gas near the center. The depletion region is displaced by 3 pc toward the NW from the AGN position, meaning some m = 1 asymmetry in the torus. The molecular torus in NGC 1672 has a different position angle from previous lower-resolution observations, and is edge-on, revealing a geometrically very thin torus (axis ratio 6.5─10), with a clear warp. This confirms that the classical model of a simple geometrically thick dusty torus is challenged by high-resolution observations. The nuclear disks appear to be clumpy, and slightly lopsided. The molecular outflow in NGC 613 is now resolved out. Well inside the sphere of influence of the black holes (BH), we are now able to determine more accurately their mass, for those Seyfert spiral galaxies, in a region of the M-sigma relation where the scatter is maximum.