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)
.
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.”
.
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
.
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