We present measurements of key protoplanetary disk properties inferred from parametric models of ALMA ^12^CO spectral line visibilities. We derived gas-disk radii, integrated fluxes, optically thick emission layers, and brightness temperature profiles for the disk population of the old (4-14Myr) Upper Scorpius star-forming region. We measured CO emission sizes for 37 disks with bright CO J=3-2 emission (S/N>10 on the integrated flux; out of the 83 disks with CO detections), finding that the median radius containing 90% of the flux is ~82au, with radii spanning from 22 up to 247au. We report a correlation between the ^12^CO brightness temperatures and stellar luminosities, with a Pearson coefficient of 0.6, which we used to prove that the ^12^CO optically thick emission layer primarily emanates from a region below the superheated dust, which is optically thin to the stellar irradiation. Moreover, we derive 33 CO emission-surface height profiles, finding a median aspect ratio of <z/r>~0.16 in a range from ~0.01 up to ~0.45 over the sample. Finally, we comment on the multiple systems in our sample, of which only some were already known. These results confirm that it is possible to derive bulk disk properties by modeling moderate-angular-resolution ALMA visibilities.