1) Pseudo-Random Number Generator with either "development" seed that worked for you or unique to each install/new game/new landfall for that little extra individuality.
2) Generate random XYZ coords, rejecting any that lie outside a significantly oblate spheroid within the cubic envelope. If a sphere, would prevent pole-clustering if you used lat/lon or ascention/declination randomising, directly. As a spheroid with maybe half the cartesian range pole-to-pole as quatorial diameter (guessing, would need tuning to be sure) it'd give you a strong 'disk' signal. I'm guessing you need only 100 or so such valid points to be useful, could be fewer, could be more, but not much either way.
3) Convert this small list to polar vectors. Ideally adding a polar tilt (before or after moving from XYZ to asc/dec, but mathematically easier before), possibly also have shifted the origin (or the points w.r.t. the origin) ⅓ or ½ the way across the equatorial disc for a further induced assymetry (best done before the tilt, before the polar conversion). You don't care about the radial factor.
3b) If you want to replicate this sky from "another planet" you just put in different shifts at that stage on your like-seeded and like-culled PRNG output.
4) Continue to use your PNRG to 'flavour' your points by hue, brightness, apparent size (I find that using a 'flat' random number for apparent radius works OK, e.g. minimal1x1 blobs (~1 pixel) as frequent as maximal 3x3 ones (~9 pixels) with the median 2x2 (~4 pixels), but a 0.5<x<1 power could remove too many "supergiants" from this sequence. You can tie hue (redish to yellowish, perhaps a tips into blue and green as a rarity, magenta should be low-probability) and brightness to size, in various ways, but we've already chucked the 'distance' from our collection of dots, so absolute and relative variations are going to be a part of this.
4b) If you
are going to reproduce "same skies, different worlds" as per (3b), then by all means pre-assign absolute values to size and brightness (with a link to hue biasing towards reasonable versions of Main Sequence astrological physics?) at PNRG positiinal-creation time and apply inverse-square scaling to each becore you lose the radial polar component, so the red supergiant close up matches the one seen from further away. But if you don't have to, it's as handy to draw off the new RNG values from the stack after the fact.
5) Now you have 5 values (position on the heavens with 2 and their dotlike appearance with the other three - HSV or converted to RGB) for maybe a 100ish points, which you then just spherically rotate and render (just those lying within the viewscreen XY, maybe 25 or so at a time depending on 'cone of vision' 3d angle) with an additional rotational matrix based upon what looks good for sidereal time and planetary inclination (different from the galactic inclination already invoked) as per the time of year) and pre-processed to filter in/out by daylight effect as 'solar time' requires. That may be enough to reveal 'familiar' constellations repeating (without a multimegabyte bitmap, and a lot less processing effort to spot-paint onto a canvas than it would be to coordinate any of the other parellax maps).
6) For additional fun, define a procedural "wave" around the galactic equator (slight illumination, dropping steeply off, serated edges by applying three or four coprime factors of the circumference to stacked sinusoidal variation to edge-limit, approximations to "dust cloud" gaps in it by darkened splodges, perhaps some slughtly-off-wave relightened splodges too) that are then processed as a further gradiated-band from sky-vector to viewport-vector by the same validation/conversion routine as a background to the 'star dots' for a slightly more processor-intensive hint that there is an ecliptic out there.
7) For
additional additiional fun, you could model other galactic 'blobs' (view origin moved
well outside the bounding/culling perimiter, with polar angle shift before point-translation, different in each case) for a small gradient-shading and a few noted point sources (low saturation, single-pixel, high brightness pinpricks) as objects to render (appropriately rotated) as per stars on the star-layer (essentially meta-stars, but be careful not to overly 'burn through' dust-patches on the layer beneath created by (6), maybe pre-filter away those that tie too close to the local galactic disc band) for the "Andromeda only a few hundred billion years or so from colliding" effect.
You know where the local Sun is, and what time of year (and what year?) it is, by this planet's standards. I bet you could code in a few
actual planetary wanderers in there, with purely emergent retrograde loops. Single-pixels (unless you want to go for an Endorian look for one of them, rather than just go Big on one particular Moon, the mechanism for which I need not describe here save that it should
not cycle the same cycle as night-time/opposite the sun unless it's a particularly tidally-locked scenario that pops out of the (PNRG?) design phase), and then there's single variiously bright pixels for comets (with additiinal faint streak, aligned to oppose the perhaps non-visible sun?).