Parody is essentially the act of mocking something in style, by using the elements or literary/dramatic techniques of an individual or genre to make fun of it. Burlesque is a direct mockery of something in particular—either a play, film, or novel.
An example of a parody would be the film series "Scary Movie". Each of these films, while they take some direct references, is imitating and exaggerating the overall style and themes that are typical of the horror genre. A burlesque example would be the film "Space Balls" which is a direct imitation and exaggeration of the Star Wars franchise, mocking specific events throughout it.
The distinction between the two is subtle, and there are other subtleties that make the two different as well.