Different citation formats are popular with different academic disciplines. Generally the forms preferred for the various disciplines are as follows:
- MLA (Modern Language Association) style (parenthetical notes): English, Foreign Languages, Linguistics.
- University of Chicago style (footnotes): Classics, Dance, Folklore, History, Music, Theatre.
When a professor asks you to include "parenthetical footnotes" he means that instead of placing the footnotes at the bottom of the page you are to place them in parentheses within the body of the text where you would normally type the note number. It is usually a good idea to use parenthetical notes for works that you site a great deal in the course of your paper (i.e., if you were writing a paper on Madame Bovary you might want to cite that primary text parenthetically). Parenthetical notes can take any number of forms: (1) they can be traditional footnote citations (above) differing only in their placement (2) they can be abbreviated, sometimes offering only the author's last name and the page number of the cited passage, or (3) they can be abbreviated to the extent of offering page numbers alone (this should only be done with ONE source in a paper).
It is generally a good idea to just ask your professor to specify which form of citation he prefers.