Growing up, some Santa gifts were wrapped in special paper. The 'big gift' (sometimes a joint gift with my brother) wasn't wrapped and my parents would set up a whole 'scene'.
For example, one year I got the entire gymnastics set for my American Girl doll, my parents dressed her up and had her 'practicing' the beam (no idea how long it too for them to get her to stay) and sprinkled wrapped candy all around it (this was before we had pets).
Any small gifts would be wrapped and placed in the stocking.
Granted, Santa usually brought a 'big' gift, and the 'parents' gave us the rest of them, which were always wrapped and then hidden around the house (which my brother and I would have to find and unwrap, in numerical order, after both had found them all).