A 'nosy parker', sometimes spelled 'nosey parker', is a person of an overly inquisitive or prying nature.
With all phrases that include a word that could conceivably be a person's name, it's natural to try to seek out the individual in question. Sometimes this search is fruitful, as with Hobson's choice and Sweet Fanny Adams for example; more often, as with 'as happy as Larry', 'Mickey Finn' etc., it is a wild goose chase.
The person most often associated with the phrase 'nosy parker' is Matthew Parker, who was Archbishop of Canterbury from 1559 to 1575. In a systematic attempt to obtain a detailed account of the qualifications and activities of the clergy he ordered several unpopular inquiries. This, and the good archbishop's impressively prominent nose, might be thought more than enough for his peers to have nicknamed him 'nosy Parker'. The problem with this story is that his peers did no such thing.
The phrase 'nosy parker' dates from the end of the 19th century. The popular Victorian novelist Mary Elizabeth Braddon edited the Belgravia Magazine at that time and 'nosey parker' appeared there in the May 1890 edition, which seems to be the first example of the phrase's use in print:
You're a asking' too many questions for me, there's too much of Mr. Nosey Parker about you, an' I'd 'ave you to know as I'm a laidee.
In the 17th century, 'nosey' was just a name for someone with a large nose (although Archbishop Parker appears to have avoided that fate). An early example is in Thomas Shelton's 1620 translation of Cervantes' Don Quixote:
The Story leaves them, to tell who was the Knight of the Glasses and his nosier Squire.
The word began to be used to mean 'inquisitive' from around the start of the 19th century. Robert Montgomery's satire The Age Reviewed, 1828, includes a reference to it in that context:
The scyophantic [sic] gang Whine through the kingdom with deceitful slang; Till nasty, nosy gabble mouth'd for hire, Puff their mean souls into Presumption's fire.
Before 'nosy parker' was coined, a 'parker' was simply a park-keeper. The opportunities for park-keepers to spy on courting couples were no doubt ample and there has been some speculation - by the lexicographer Eric Partridge and others - that this may be the source of the term. Such spying probably did go on, and it is now such a popular pastime as to have been given a name - dogging, but there's no real evidence of a link between furtive park-keepers and 'nosy parkers'.
Was the first Nosy Parker a real person and, if so, who? We don't know. What is certain is that the rather hopeful guess that it was Parker the hawk-nosed butler from the 1960s Thunderbirds puppet series can be ruled out.
It is much more likely that the expression alludes to overly inquisitive people who 'stick their noses in' other people's business. The same allusion was probably called on with the coining of the more recent and graphic New Zealand/Australian phrase with the same meaning - 'sticky beak'. Where parker comes into it is anyone's guess.