"That’s the Duke’s town palace, built by Palladio."[1]
The old man is referring not to the villa at Vicenza where the story is to take place but to the Duke's other residence across the valley.
Andrea Palladio, Andrea di Pietro della Gondola, (1508 - 1580) is an Italian architect. He is regarded as the greatest architect of northern Italy around the 16th century and is famous for his palaces and villas.[2] If one of these villas is the property of the present Duke and possibly the past Duke Ercole II, it is a demonstration of power and affluence for to have something built by Palladio would have been a mark of wealth and influence.
[1] Wharton 1901, p. 2
[2] see Britannica 7