Satire and parody are not exactly the same thing, but they are related. A parody, which The Importance of Being Earnest certainly is, takes on some established form of art (in this case the theatrical romantic comedy) and makes fun of it. This can be entirely for fun, or it can have social or political implications (Earnest has both.) A satire is not necessarily an imitation of some other form -- it is merely a form of art which critiques things wrong with the human condition. This can be political or social or cultural, but a satire doesn't specifically make fun of a form of art -- it makes fun of a state of affairs or a social norm. So when the manners of the English aristocracy are made fun of, it's satire, and when the crazy conventions of turn-of-the-century romantic comedy are skewered, it's parody. There are elements of satire in Earnest, especially of the English upper classes, but play is more of a parody.
So what's satiric rather than parodic in The Impotance of Being Earnest? The imaginary friend, Bunbury, is definitely a satirical, homosexual reference. Wilde, persecuted by the law and society for his sexual orientation, was only to happy to put in a joke that only certain people would understand, as a sort of "shout-out" to the English homosexual community.
The absent-minded, somewhat silly vicar Mr. Chasuble is another, somewhat gentle, object of satire. He is an element of parody, too, but Wilde is certainly making fun of the Church of England in this character's scatterbrained patter.
Lady Bracknell, the voice of propriety and the old generation, is definitely an object of satire (of parody, too, because her type of character is a staple in romantic comedies). Her illogical reasons for not allowing the young people to marry, and her somewhat mercenary motives for things, are definitely a critique of high society of that time.