In the Union army in the US Civil War, the higher the rank the more likely someone would be to have a full beard, at least in camp. On the march everyone might have to do without shaving for days, weeks, or months. But almost every single general was photgraphed with a full beard, even in studio portraits where they had chances to shave if they wanted.
Lower ranking officers and enlisted men were more likely to display only partial facial hair.
In the British army around 1900 there were regulations about facial hair. Enlisted men (except for really young ones) were required to grow mustaches.
I presume officers were free to grow as much or as little facial hair as they wanted, which would mean that some officers had full beards.
So maybe at the time of "Errand of Mercy" facial hair was a sign of rank in the Klingon forces. So maybe the enlisted mean were shaven and only some officers were allowed to grow beards.
In one-hour drama world, I've seen a hospital show make a big point of how doctors in training wear short white coats, while the long coats are reserved for their bosses, the attending physicians. And there's a current cop show making hay out of rookies having to wear long-sleeved uniforms, while the short sleeve version is a mark of status for fully-trained cops who've made the cut. The upper echelons in a hierarchy often figure out ways to ensure a display of status.