Can New Horizons be used to measure the distance to Betelgeuse (despite its fickle photocenter)? The New Horizons spacecraft is almost 50 AU from Earth and near-simultaneous images taken by its telescope and one on Earth have demonstrated parallax without relying on the movement of Earth in it's orbit.
The simultaneity is important because measurements of Betelgeuse are confounded by its fickle photocenter†. It's apparent disk is lumpy and nonuniform; the &quot;center of gravity&quot; of intensity of its apparent disk moves around relative to its true center of mass. Simultaneous imaging from two widely separated positions using the same spectral bandpass could be used to try to address that problem.
Could this be done given what's known about its Long Range Reconnaissance Imager or LORRI and considering that the spacecraft has plenty of time on its hands?
Recent headlines saying that Betelgeuse is much closer than we thought point to Standing on the Shoulders of Giants: New Mass and Distance Estimates for Betelgeuse through Combined Evolutionary, Asteroseismic, and Hydrodynamic Simulations with MESA which proposes in the abstract

...a new, independent distance estimate of ${168}_{-15}^{+27}$ pc and a parallax of $\pi ={5.95}_{-0.85}^{+0.58}$ mas, in good agreement with Hipparcos but less so with recent radio measurements.

based on extensive modeling and photometric data.
What's truly remarkable is that I just copy/pasted that blurb and it self-formatted in MathJax, yay!
New Horizons is currently at X, Y, Z = 14.11, -46.42, 1.67 (AU) relative to the solar system barycenter in J2000.0 with ecliptic as reference plane using JPL's Horizons. I'm hoping that's orthogonal to Betelgeuse.

†fickle photocenter You can try to say that three times fast, but whatever you do, don't say the star's name three times!

Related:

What will it finally take to accurately measure the distance to Betelgeuse?
What equipment and techniques were used to study Betelgeuse&#39;s diameter in 1920?
How often are radio telescopes used to measure parallax? When was this first done? Currently unanswered.
Was a “spare LORRI” camera available to take matching image on Earth the same time New Horizons took the parallax image from the Kuiper belt? No conclusive answer yet, I really thought that there was one, and it might come in handy for this.
In principle yes, in practice no. The telescope is good enough, but the CCD camera will saturate, preventing a good positional measurement.
Salient facts. The parallax to Betelgeuse as seen between Earth and New Horizons will be about 250 mas. The current distance uncertainty is ~20%, so to better that, the position of Betelegeuse measured by New Horizons would need to be more precise than ~25 mas.
Looking at the LORRI camera, it has a plate scale of 4.95 microradians/pixel and a point spread function with FWHM of 1.8 pixels, which equals 1.84 arcsec (Cheng et al. 2007).
A useful rule of thumb is that your positional accuracy in a resolved image is given approximately by
$$ \Delta \theta \sim \frac{ {\rm FWHM}}{{\rm SNR}}\ ,$$
where SNR is the signal-to-noise ratio of the image. Using this, it suggests that if one could get an image of Betelgeuse with a SNR of around 100, then you should be able to centroid the position to better than 25 mas using the New Horizons data. Combining this with a simultaneous Earth-based image (which one presumes would be much more precise), then the parallax could be determined to better than 10%.
The problem appears to be the brightness of Betelgeuse. Cheng et al. also give a photometric calibration formula for the number of analogue to digital units per second, $S$, recorded by the camera for a point source with a given $V$ magnitude.
$$V = -2.5 \log S + 18.94 + 0.4\ ,$$
where the extra $0.4$ is probably appropriate for a star of Betelgeuse's red colour.
Using this formulae we see that $S = 2.16\times 10^8$ ADU/s, assuming $V\sim 1.5$ for Betelgeuse.
The CCD camera on LORRI only returns 12 bits and has a frame transfer time of 13 ms, which means that 13 ms is about the smallest exposure time you could use and hope to get accurate positional information (Cheng et al. 2007), though in principle a 1 ms exposure time could be done. Even in 1 ms you would get $2.16\times 10^5$ ADU from Betelgeuse spread over perhaps 4 pixels. Given that a 12 bit ADC saturates at 4096 ADU, then Betelgeuse is more than 10 times too bright and will produce a completely saturated image whose centroid cannot be determined with any precision. If it were possible to remotely reset the gain on the CCD (currently 22 electrons/ADU) to something much larger, then maybe it would be possible.