Good question. It all depends at what time you are going in January. If it is before the 5th, i believe the Christmas Fantasy Parade will still be making the 'rounds. If it is after that, i am not sure. I think the Celebration parade starts after March because of the repaving project on Main Street. Hope this helps!!!
There will not be any parades in Disneyland starting January 5th (through most of March). Fantasmic, however, will probably have a ~7:30 showing on weekdays, and normal showings on weekends. California Adventure will have Pixar Play Parade, but it will be only running 5-days per week.