Merriam-Webster's Eleventh Collegiate Dictionary (2003) reports six distinct senses of the prefix para- (or par-):
1: beside: alongside of: beyond: aside from <parathyroid> 2a: closely related to <paraldehyde> b: involving substitution at or characterized by two opposite positions in the benzene ring that are separated by two carbon atoms <paradichlorobenzene> 3a: faulty: abnormal <paresthesia> b: associated in a subsidiary or accessory capacity <paramedical> c: closely resembling: almost <paratyphoid>.
In addition, the Eleventh Collegiate lists one combining form of para-:
para- comb form [parachute]: parachute
But it doesn't list another combining form that is evident from the opening portion of the dictionary's entry for Paralympics:
Paralympics n pl [paraplegic + Olympics]
Note: As Avon points out in a comment below, the International Paralympic Committee, which organizes the Paralympic Games, now presents the word paralympic as referring not to paraplegic but to parallel. The Wikipedia article on these games explains the change:
Although the name was originally coined as a portmanteau combining "paraplegic" (due to its origins as games for people with spinal injuries) and "Olympic," the inclusion of other disability groups meant that this was no longer considered very accurate. The present formal explanation for the name is that it derives from the Greek preposition παρά, pará ("beside" or "alongside") and thus refers to a competition held in parallel with the Olympic Games.
I couldn't find any non-proper nouns in the dictionary that use para- as a combining form with the meaning "paraplegic," but a Google search reveals that some parts of Alberta and Yukon Territory in Western Canada use the term pararamp to refer to wheelchair-friendly sidewalk-to-street ramps on city streets. Since these are not alternatives to some sort of normal ramp (normally, sidewalks end in curbs that drop off to the street several inches below), I don't think that any of the six senses of para- that the Eleventh Collegiate identifies apply here. If that's correct, then the para- in pararamp would seem to be short for paraplegic or paralytic.
The same cannot be said for paratransit and paratransport, which refer to services that are ancillary to the standard transit and transport options in the municipalities that offer them. Of course, many people who have difficulty with standard mass-transit options are neither paraplegic nor wheelchair-bound; but I suspect that the coiners of pararamp may have been influenced by their awareness that paratransit and paratransport do offer special services for people in wheelchairs.
In any event, as FumbleFingers says, parabrake and paraglider (like paratrooper) fall into the combining form of para- that signifies 'parachute." A final point: The words parasol, parachute, and paravane (referring to "a torpedo-shaped protective device with serrate teeth in its forward end used underwater by a ship in mined areas to sever the moorings of mines") use para- in the sense of "to ward off or to shield"; in the context of parachute, the Eleventh Collegiate says, chute means "fall."
Andrew Leach's answer has the OED's first quotations [parenthetically in 1884, and] in 1890. Their first quotation for "in good nick" is The English dialect dictionary from 1905.
Australia, 1880s
I found earlier uses in the Trove archive of Australian newspapers, the earliest in The Referee (Sydney, NSW, Thursday 13 January 1887):
Hutchens and Samuels.
(By "Shoespike.")
Next Monday Hutchens will run his first
match in Australia. Malone's was to have
been the first, but the aboriginal party
were found willing to risk a century, and
a match was quickly made. Samuels has
not had much time for preparation, but is
quietly doing work on the Agricultural
Ground. He looks if anything fine, and
not so strong and in such good "nick" as
when he won the Botany. As an aboriginal
Samuels is a first-rate runner, and about
the best of them. I question, however, if
he is class enought to stretch the world's
champion and anticipate Hutchens to
win comfortably. I may add I do not
expect even time to be broke.
It was used in other Australian newspapers in the late-1880s to describe sporting participants: wrestlers, racehorses, footballers, boxers rowers.
New Zealand, 1870s
However, it can be found earlier in New Zealand's archive of newspapers, Papers Past, and again in a sporting context. First in Sporting Notes by "Sinbad" in The Press (Volume XXIX, Issue 3973, 18 April 1878, Page 3), describing racehorses:
York, the representative of the Bay stable, is big enough and strong enough. Those who ought to know say he has plenty of pace, and will certainly be there or thereabouts at the finish. He is without doubt in good nick, and will have a good man on his back, so I think he will run into a place, and if either Natator or Merlin are out of it he may be labelled dangerous.
(The article also uses the similar phrase in good form.) In good nick shows up in many other editions of The Press and also The Obago Witness in the late-1870s, all applied to racehorses.
An origin?
Another meaning of the noun nick dates from 1824 and, according to the OED:
10. An instance of cross-breeding, esp. one which produces offspring of high quality. Cf. nick v.2 7b.
You could say of animals or racehorses, as in this from an 1870 Australian newspaper:
It is possible,
however, as the mare is a daughter of Melbourne, that
Stockowner may prove a good nick.
From the same article, as a verb:
I see that a certain sire and dam "nick" well, no
matter how wrong it may be for them to do so, as far as
the relationship of their families is concerned, I prefer
to trust to their progeny, rather than to thoso bred on a
correct theory without practical results.
So perhaps as the term for successfully crossed animals, specifically racehorses, was applied to racehorses generally in good form. This was then used for sportsmen in general before being used for anything in good condition, or conversely, as "in poor nick" for something in bad condition or form.
Best Answer
Etymonline to the rescue:
The full OED entry for the word mentioned by Etymonline is much too long to fully cite here, but includes the following material pertinent to the current discussion:
The earliest provided citation dates from a 1656 work by antiquary and lexicographer Thomas Blount in his Glossographia; or, A dictionary interpreting all such hard words, whether Hebrew, Greek or Latin... as are now used in our refined English tongue (1ˢᵗ edition, 1656, London), where he writes:
This was quickly followed in 1667 by Jeremy Taylor, Church of Ireland Bishop of Down and Connor, in the 4ᵗʰ edition of his work The great exemplar of sanctity and holy life (1ˢᵗ edition 1649, 4ᵗʰ edition 1667), where he writes at ɪ. vi. §22. 160:
So, it seems that it is based on a misspelling of a Greek word — but that would hardly give one reason to form the plural in Greek starting from the Latin(ized) form.
According to both Merriam-Webster and the OED alike, the plural can be either syllabi /ˈsɪləbaɪ/ or syllabuses /ˈsɪləbəsɪz/, but with Etymonline’s “no real word” verdict, I’d go for the latter.