At our house Santa fills up their stockings and brings one or two other presents, the rest are from mom and dad. That's how DH and I grew up, and it makes sense to me that mom and dad would give most of the gifts because Santa has to give gifts to all the kids around the world, so he doesn't have room in his sleigh for more than that per child.
I had to add, Santa wraps the gifts at our house before he leaves them.