In the West, before the twentieth century this really didn't take place, because the notion of radically different styles representing not just artistic individualism but different or opposing cultural arguments--different ideas of what art is--did not yet exist. Stylistic parody in painting would only become possible with the development of the idea of the avant-garde and the fracturing of culture that it implied, over the course of the 19th century. But there are precedents.

There is the case of Titian's caricature of the Laocoon, an important classical sculpture group unearthed in Rome in 1506; Titian reduces its figures to monkeys in what is perhaps a jab at contemporary idolators of the sculpture. But this is not a stylistic satire but rather a substitution of subject matter.

More common were pastiches. A recent book has argued that Chardin, traditionally regarded as a kind of 18th-century realist avant la lettre, faithfully documenting bourgeois life, actually was engaged in pastiche of baroque Dutch still life painting in his genre scenes. This is a kind of knowing use of an older style, informed by a playful, aristocratic aesthetic of galanterie. This is not the same as parody or satire, rather is a game of references. Of course there were paintings that satirized their subject matter, such as Girodet's attack on the actress Mlle Lange in this 1799 painting that symbolically called her a prostitute, vengeance for her rejection of his first portrait of her.

Things began to shift in the 19th century as one saw flourish satirical prints (not paintings) representing paintings shown in the Salons, as /u/emotional_alien rightly suggests. It became possible to satirize a painting on the basis of its style only once multiple, competing stylistic options emerged in art exhibitions. This began around the time of Daumier and the rise of stylistic avant-gardes, from Realism, through Impressionism, various post-Impressionisms etc. This fracturing didn't exist before the 19th century, and it paved the way for the stylistic multiplicity, and stylistic parody that went with it, of the 20th century.