A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2020 May 24, 20:17 -0700
The parallax in azimuth is zero except for a minute offset caused by the slight oblateness of the Earth. That's one part in 300. The parallax of the Moon is about a degree so to order of magnitude you can expect for the shift in the Moon's angular position due to the slight change in parallax caused by oblateness to be 60 minutes of arc divided by 300 or 0.2', some in altitude, some in azimuth.
This small modification to parallax matters slightly for lunars, though it does not need to be portioned out in azimuth and altitude. Historically it was usually ignored, and there was little to worry about. It's no real problem to include this correction today if you're doing the work all in an app, and my web app has included this small correction for many years. You can experiment with it in the web app by clicking the check box to ignore it. Clear "before" and "after" and see how much change there is.
I can't think of any other case off the top of my head where this correction would matter. One could observe the leading limb of the Moon for an azimuth, but would anyone bother? Has this ever been considered "worth the effort" outside of a theoretical exercise?