What does "Oxford Classic" mean in this context?

From a letter from Charles Darwin to J.D Hooker in 1874 (Source):

It is, if you ever attend the Balloting meetings of Athenæum Club, to attend the first one early in February & vote for my nephew, Henry Parker, & ask anyone whom you can influence. He is a fellow of Oriel College Oxford, & a most able & accomplished man, & I assure you in every way fitted to be a Member.

Here is one definition of classic:

An artist, author, or work generally considered to be of the highest rank or excellence, especially one of enduring significance.

(The Free Dictionary)

Darwin referred to Henry Parker as "a most able & accomplished man". This is in line with the above definition of classic. Henry Parker also studied at Oxford, which would then make him an "Oxford Classic". Similarly, a classic from Cambridge would be called a "Cambridge classic".

A classic is (or was, back in Darwin’s day) a student of Greek and Latin literature, or a classical scholar:

classic, adj. and n.
B. n.
2. b. A student of Greek and Latin literature, a classical scholar. Now rare.
_{1805   H. K. WHITE Let. 18 Oct. in Remains (1807) I. 179   I find I am a respectable classic.
1823   C. LAMB in London Mag. May 534/2   A fine classic, and a youth of promise.
1895   J. M. FALKNER Lost Stradivarius Epil. 260   He had always been an excellent scholar, and a classic of more than ordinary ability.
1907   E. M. FORSTER Longest Journey xvi. 190   He was not a good classic, but good enough to take the Lower Fifth.
1952   R. MACAULAY Let. 2 May (1961) 309   So many brilliant classics can’t do Maths, and vice versa.
Source: Oxford English Dictionary (login required)}

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Although I don’t see a reference to a specific degree for Henry Parker, he is listed in Darwin: A Companion as a “Classical Fellow of Oriel College, Oxford,” which would seem to qualify him as a classical scholar at Oxford — an Oxford Classic:

Parker, Henry [II], 1827-92. 2d child of Henry Parker [I] and Marianne Darwin. CD’s nephew. Classical Fellow of Oriel College, Oxford. 1862 Reviewed Orchids in Saturday Rev. 1862 Dec. 29 Visited Down House. Unmarried.

English Men of Science mentions Parker as a “fellow of University College, Oxford, classical scholar and chemist.”

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Here are more examples of classic meaning a student of Greek and Latin literature or a classical scholar:

If as Head Master I wanted some piece of work done, I should feel safer if I entrusted it to a mathematician or to an Oxford classic: fairly safe if my classic came from Cambridge: but profoundly uneasy if I had to give it to a scientist.
_{— Things Ancient and Modern, Cyril Alington, 1936}

A Cambridge mathematician and a Cambridge classic are the two new judges whose appointment was gazetted on Tuesday.
_{— The Cambridge Review, Volume 28, 1907}

An Oxford classic* has drawn a comparison of the observations made by Aristotle and Shakespeare respectively on the passions, habits, and institutions of mankind . . .
_{— A Few Stray Thoughts Upon Shakespeare, Sir Thomas Howel, 1867
*Referring to Joseph Esmond Riddle, B.A. Classics}

A pure classic is no more necessarily fit to teach English History than a mathematician is to teach Iambics!¹
_{—Clear Thinking, Or, An Englishman’s Creed, Leslie Cecil Smith, 1914
1 Although the Oxford classic has at least some general training in historical method.}

The poet Wordsworth was not, like others of his family, a Cambridge Classic, but he spent much of his time, at Cambridge and afterwards, in desultory reading of the Classics . . .
_{— The Scot’s Magazine Volume 5 1890}

Nay, nay, we do not want a first-rate classic, or mathematician, or musician, or geologist, or anything of that kind. All we need is some one who will be kind to the books.
_{— The Parochial (Oxford parochial) magazine [afterw.] The Oxford magazine and Church advocate, Volume 3, 1863}

Of course a mind which has undergone systematic training of any kind has so far an advantage over an untrained mind in mastering any subject whatever, but the qualities of mind which make a man a good classic or mathematician arrive at maturity earlier than those which make him a philosopher.
_{— The Student’s Guide to the University of Cambridge. By various writers. Edited by J. R. Seeley, University of Cambridge, Sir John Robert Seeley, 1866}

He was a man of the highest order of human intellect — an accomplished physician, a learned writer on sacred subjects, a firstrate classic, mathematician , and musician.
_{— Rambles about Bath, and its neighbourhood, James Tunstall, 1847}

More at Google Books “classic” “mathematician” — 19th century

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_{Just a couple of notes regarding some other answers here . . .}

_{1. One imagines that, back in the day, Darwin would not have used the term Oxford Classic to mean, simply, an awesome student at Oxford.}

_{2. Parker’s listing as a “fine art specialist” at Epsilon is due to his appearance in A supplement to Allibone’s Critical dictionary of English literature and British and American authors (CDEL) for his book The Nature of Fine Arts, published in 1885, when he was 58 years old (in other words, it doesn’t refer to his college major):}

_{Source: A supplement to Allibone’s Critical dictionary of English literature and British and American authors}

Merriam Webster

classic (noun):

a typical or perfect example

Hence, an Oxford Classic is a perfect example of an Oxford-educated person, regardless of their field of excellence. The term might apply to those educated in the Classics, the Arts, or any other field of study that was current in Oxford in Darwin's time.

Whether one regards classic as a noun or an adjectival noun does not seem to matter.

The noun classic has various subsenses. Merriam-Webster includes the relevant (though doubtless dated):

classic [noun]

...

2a: a work of enduring excellence

• His manual of biology has become a classic among scientists.

also : its author

• He had already become a classic many years before his death.

b: an authoritative source

If the noun were 'classicist', the field of study would certainly be the classics, but here there's doubt. 'A published Oxford scholar [at whatever level]' is more likely.

...........

In fact, Henry Parker was perhaps a fine arts scholar rather than a classicist:

• Henry Parker (1827–92) Fine art specialist. Scholar, University College, Oxford, 1846–51; fellow of Oriel College, Oxford, 1851–85. Son of CD’s sister, Marianne Parker. Sources Alum. Oxon.

[Epsilon.ac.uk]

[I've edited in line with Tinfoil Hat's research, attesting that Parker was indeed a classicist (as well as a 'fine art specialist').