UPDATE — March 29th: Activision has announced that Awakening DLC will in fact be coming to last-gen platforms after all. Awakening will be available on PS3 on April 5th.
In the new Nikoli Memories video, at the end, a new teaser image for the poster of Black Ops 3 Awakening DLC has revealed that this DLC pack will not be available on PS3 or Xbox 360.
Back in October, Activision confirmed that the Season Pass itself would not be available on last-gen, but they said that individual DLC packs might be available. As of now, that looks like it will not be happening.