We present the first results from the Southern Hemisphere Parallax Interferometric Radio Astrometry Legacy Survey: 10{mu}as accurate parallaxes and proper motions for two Southern Hemisphere 6.7GHz methanol masers obtained using the inverse MultiView calibration method. Using an array of radio telescopes in Australia and New Zealand, we measured trigonometric parallaxes and proper motions for the masers associated with the star formation region G232.62+00.99 of {pi}=0.610+/-0.011mas, {mu}_x_=-2.266+/-0.021mas/yr, and {mu}_y_=2.249+/-0.049mas/yr, which implies its distance to be d=1.637+/-0.029kpc. These measurements represent an improvement in accuracy by more than a factor of 3 over the previous measurements obtained through Very Long Baseline Array observations of the 12GHz methanol masers associated with this region. We also measure the trigonometric parallax and proper motion for G323.74-00.26 as {pi}=0.364+/-0.009mas, {mu}_x_=-3.239+/-0.025mas/yr, and {mu}_y_=-3.976+/-0.039mas/yr, which implies a distance of d=2.747+/-0.068kpc. These are the most accurate measurements of trigonometric parallax obtained for 6.7GHz class II methanol masers to date. We confirm that G232.62+00.99 is in the Local Arm and find that G323.74-00.26 is in the Scutum-Centaurus arm. We also investigate the structure and internal dynamics of both masers.