How to calculate parallactic angle from a fixed alt-az position?
I found an equation on this website,

$\require{cancel} \cancel{\sin{\theta_p}=\sin{\theta_{az}} \cos{\theta_{lat}}/ \cos{\delta}}$,

with $\theta_p$ being the parllactic angle, $\theta_{az}$ being the azimuth, $\theta_{lat}$ being the latitude of the observing stations, and $\delta$ being the declination.  The website cites this equation as being from "Spherical astronomy, Small pg. 49."  I am unable to find such a book (based on my searches, I'm wondering if the correct reference might actually be "Textbook on Spherical Astronomy, Smart", which I do not have ready access to), but in my experience so far this equation has been correct.

The above equation, my original answer, seems to provide something similar to the parallactic angle (something like the complement in each quadrant, with an extra negative sign, but I haven't investigated in detail).  However, when testing against values gotten from astropy, it did not match the astropy values.

Instead, I found the following, taken from Astronomical Algorithms by Jean Meeus, which does match the astropy values:

$\theta_p = \text{atan2}( \sin{\theta_H}, \tan{\theta_{lat}}\cos{\delta} - \sin{\delta}\cos{\theta_H}),$

with $\theta_H$ being the hour angle, calculated via

$\theta_H = \text{atan2}( \sin{\theta_{az,mod}}, \cos{\theta_{az,mod}} \sin{\theta_{lat}} + \tan{\theta_{el}}  \cos{ \theta_{lat} } ),$

Where $\theta_{el}$ is the elevation and $\theta_{az,mod}$ is the azimuth, but defined differently from $\theta_{az}$ as used originally and elsewhere in my answer.  The zero-point $\theta_{az}$ is due north, increasing clockwise, and the zero-point $\theta_{az,mod}$ is due south, increasing clockwise.

Declination can be gotten from latitude, elevation, and azimuth:

$ \sin{\delta} = \sin{\theta_{el}} \sin{ \theta_{lat} } + \cos{ \theta_{el} } \cos{ \theta_{lat} }  \cos{\theta_{az}} $,

Thus indeed, given an alt-az (or az-el depending on your preferred terminology) position and the latitude of the observing station, it is unnecessary to use RA or an independently determined hour angle in a calculation of parallactic angle. I say "independently determined hour angle" meaning that although hour angle is used in the calculation, it can be determined entirely from az/el and latitude, so one does not need to provide an hour angle for this calculation to work.