"Could this be used to make a more accurate parallax measurement ?"
For 'close' stuff, such as TNOs, the Kuiper Belt & the inner Oort-cloud, surely, but I don't think the camera's optics give enough resolution for full-on stellar Astrometry...
New Horizons hasn't gotten beyond the known outer planets yet. Until it gets to the distance of Eris' aphelion, 97.651 AU, it's still traveling among the outer (geophysically defined) planets.
Nik_2213
Feb 9, 2018"... spacecraft is healthy and is currently in hibernation."
So, what is it doing ??