  Recently, GRAVITY onboard the Very Large Telescope Interferometer (VLTI)
first spatially resolved the structure of the quasar 3C 273 with an
unprecedented resolution of $\sim 10\mu$as. A new method of measuring parallax
distances has been successfully applied to the quasar through joint analysis of
spectroastrometry (SA) and reverberation mapping (RM) observation of its broad
line region (BLR). The uncertainty of this SA and RM (SARM) measurement is
about $16\%$ from real data, showing its great potential as a powerful tool for
precision cosmology. In this paper, we carry out detailed analyses of mock data
to study impacts of data qualities of SA observations on distance measurements
and establish a quantitative relationship between statistical uncertainties of
distances and relative errors of differential phases. We employ a circular disk
model of BLR for the SARM analysis. We show that SARM analyses of observations
generally generate reliable quasar distances, even for relatively poor SA
measurements with error bars of $40\%$ at peaks of phases. Inclinations and
opening angles of BLRs are the major parameters governing distance
uncertainties. It is found that BLRs with inclinations $\gtrsim 10^{\circ}$ and
opening angles $\lesssim 40^{\circ}$ are the most reliable regimes from SARM
analysis for distance measurements. Through analysis of a mock sample of AGNs
generated by quasar luminosity functions, we find that if the GRAVITY/GRAVITY+
can achieve a phase error of $0.1^{\circ}$ per baseline for targets with
magnitudes $K \lesssim 11.5$, the SARM campaign can constrain $H_0$ to an
uncertainty of $2\%$ by observing $60$ targets.
