Is there any preference to using "Schönes Wochenende" or "Schönen Wochenende" as a parting statement? Are they both allowed, or is there a preference for one over the other? Different people have told me that one or the other is OK, but that the other is wrong. This leaves me highly confused, and I often find myself going back and forth between the two.
Yes, they are both gramatically allowed in the right context, and there is a difference between them regarding the use of the neuter "[das] Wochenende".
"Schönes Wochenende" is singular Nominative or Accusative, like "[ein] schönes Wochenende".
"Schönen Wochenende" is singular Dative or Genitive, and you'd probably attach a definite article or prepositional phrase to it like "dem/des/am schönen Wochenende".
If, for example, you want to wish a friend a nice weekend, you would say, "Schönes Wochenende!" as if to say "Ich wünsche dir ein schönes Wochenende!"
Here's a link I found that gives some different examples and translations of this phrase: http://www.linguee.de/deutsch-englisch/uebersetzung/eine+sch%F6nen+wochenende.html
While being German my Grammar is a bit rusty if put to real test, however, in a parting statement you never say "schönen Wochenende".
You would only use "schönen Wochenende" in a descriptive context, like - on a nice weekened we could go for a picknick - "An einem schönen Wochenende könnten wir...".
Generally accepted on parting, casual "Schönes Wochenende!" or more formal "Ich wünsche Dir/Ihnen ein schönes Wochenende".