Can a essay being addressed to one specific audience?For example, if I want to address it to my dad, can write "Do you remember, dad? That time our whole family travelled to New YorK?"
Given that an essay is, simply, a piece of writing on a particular subject, I could not find any reason as to why it could not be written to a specific audience.
For example, if you are writing an essay to explain the Naturalistic period, you would want to know who your audience is so that you could customize the language so as to elevate it (when your audience is a board of professors) or lower it (when your audience is high school students).
Any essay is best produced when the author keeps the audience in mind. Using terminology which is over the head of your audience will always force them to become confused or turn off completely. If writing about medical procedures for someone getting ready to have surgery, one would not want to use the medical terms used by doctors. Instead, they would use layman's terms to insure understanding for the audience.
As for writing an essay and addressing it to your father, based upon the answer provided above, there should be no problem. The only thing that you would need to watch would be if the assignment requires you to follow the general rule of not using the pronoun "I" in formal writings.