I think that Peter Shor's analysis is very likely the correct one. In support of it, I offer two early instances of "low key" in the sense of low vocal pitch and two early instances of figurative use of "low key." All are drawn from the Elephind newspaper database.
The earliest literal use of "low key" that the Elephind search turns up is from "Interesting Story Extracted from the 'Pioneers'," in the [Vincennes, Indiana] Western Sun (April 12, 1823):
But when aroused by this cry from Louisa, Miss Temple turned, she saw the dog with his eyes keenly set on some distant object, with his head bent near the ground, and his hair actually rising on his body, either through fright or anger. It was most probably the latter for he was growling in a low key and occasionally showing his teeth in a manner that would have terrified his mistress, had she not so well known his good qualities.
Although "low key" might refer to a deep (or bass) pitch—treating the growl as a note of music—I think that it more probably signifies low in loudness—a soft but ominous growl.
And from "Parliamentary Portraits," in the Sydney Gazette and New South Wales Advertiser (November 9, 1837):
He [Mr. Aglionby] speaks with singular rapidity : I am not sure whether he does not speak a greater quantity in a given time than any member in the house. No reporter could, if he wished, follow the hon. member through his speeches; that, however, for the reasons I have already given, is never attempted. His voice is not strong, but it is clear. It is easier to hear than to follow him. He never raises his voice: he continues in the same low key throughout.
Here, "low key" may refer both to loudness and to pitch, as the latter tends to go up when the former increases.
The earliest figurative use of "low key" is from "A Musical Definition," in the [Leesburg, Virginia] Genius of Liberty (October 23, 1830):
A gentleman, whose real name was George Sharp, but who generally went by the appellation, amongst his musical friends, of G sharp, on entering the company, and looking rather dull, a common friend observed that Mr. G. Sharp, was rather on a low key, that evening.—"O," replied a lady, with a good deal of naivette, "every body knows that G sharp is A flat."
The joke turns on the naive lady's literal interpretation of "low key"—but that interpretation wouldn't be a source of mirth if the common friend hadn't meant "low key" in the sense of "looking rather dull."
From "Jenny Lind in the Concert-Room," in the [Hobart, Tasmania] Courier (December 18, 1847):
The pieces set down for the prima donna were " Quando lascini la Normandia," from "Roberto il Diavolo," " Su l'aria," from "Il Nozze di Figaro;" lastly, the simplest, but most alluring entry in the whole list, was "Swedish melodies—by desire." By the desire of whom we know not; but are quite sure that, whoever they may be, there is not one of the audience who is not ready to return them his very hearty thanks for having suggested the enjoyment these melodies afforded. Those who have witnessed Jenny Lind's dramatic powers pitched, in all probability, their expectations in a low key, when they went to hear her with the stage excitements and opportunities of dramatic expression withdrawn, as they are at a day-light concert. If they did they were they were wise, for during the first performance of the first piece the singer was labouring under a degree of nervousness which extended its effects to her voice. The encore, however, reassured her, and she produced more effect than could have been anticipated.
The sense of "low key" here clearly involves restraint rather than literal quietness, but the writer explicitly frames it as a form of metaphorical voice: "pitched ... their expectations in a low key."
Also of possible interest is this instance from Charles Dickens, The Life and Adventures of Martin Chuzzlewit (1844):
She [Mrs. Gamp] continued to sidle at Mr. Chuffey with looks of sharp hostility, and to defy him with many other ironical remarks, uttered in that low key which commonly denotes suppressed indignation; until the entrance of the tea-board, and a request from Mrs. Jonas that she would make tea at a side-table for the party that had unexpectedly assembled, restored her to herself.
This example is significant because it associates "low key" not simply with low vocal pitch but with the suppression or restraint of the character's natural inclination to express her indignation loudly. As Peter Shor observes, it would be no great leap for this notion of "low key" in the sense of restrained or subdued to emerge from an earlier, more literal use of "low key" in the sense of quiet or low pitched.
Best Answer
Within linguistics, the terminology for parts of words varies according to theoretical perspective. Commonly, though, they are called morphemes.
Morphemes are frequently divided into various kinds, most basically, roots versus affixes. Roots are “open class” items, like objects (cat, chair, paper) or actions (run, arrive, fall), whereas affixes are “close class” items (such as the verbal endings ing, s, ed). That is to say, within a language, the stock of roots is large and expandable (when you encounter a new animal or invent a new gadget, you can coin a new root), whereas the set of affixes is small(er) and generally fixed (people rarely invent new verbal affixes).
Returning to your question, you can refer to para, sol, jay, and walk all as morphemes. Of these, para is an affix, and sol, jay, and walk are roots. Being a combination of two roots, jaywalk is said to be a compound (and, given that you can’t derive its meaning from the meanings of its constituent morphemes, it is said to be semantically opaque, or idiomatic). The root sol (also found in solar, solarium, solstice) does not occur in isolation (as a freestanding word sol), unlike jay and walk. So, it is called a bound root (or a cranberry morpheme, after the cran or cranberry, which also does not occur in isolation); jay and walk are free roots.