The personification of inquisitiveness: coll.; since mid-19th c. The OED dates the former at 1851, and parker, a rabbit living in a park, at 1846. Ramesey Spencer very shrewdly comments: 'The Great Exhibition took place in 1851, in Hyde Park, drawing huge crowds of inquiring visitors of all classes, so it may also have provided unusually ample opportunities for Peeping Toms and eavesdroppers in the purlieus'. That, clearly, supplies a vastly more probable origin than the various folklorish theories we have seen.
(A Dictionary of Slang and Unconventional English)