According to a 22 September 2010 interview with the designer of the current website, Jennifer Over, the Paris Review logotype is not something you'll be able to download:
We also really wanted to embrace certain design elements of the print magazine, like the hand-rendered logotype, some of the mid-century-modern typography and the frontispiece illustration by William Pène du Bois.
The same question was asked on WTF in 2010: it remains "unsolved".
Late reply as it's interesting. While custom, this looks a lot like Fournier, the Roman du Roi or some of the other fonts of the eighteenth century. The tell is the curled leg of the 'R', which is an eighteenth/late-seventeeth-century feature, and the narrow, almost monoline serifs, although the 'h' with its leg folding inwards is more sixteenth. Fournier has been digitised by Monotype, Typofonderie and Joshua Darden, so you have a bit of choice. Fleischmann's types are also in the same mood.