No, just the kids. But they do get something from H and me. My sister asked my mom one year, after opening all her gifts from "Santa" "where's my gift from you and daddy?" whoops! lol
My kid is too little to understand just yet. I think in our house Santa will fill all the stockings, but only the kid (or kids) will have Santa presents under the tree.
A whole cast of characters wrapped presents for me as a kid. Santa, Charlie Brown, Linus, Snoopy, Rudolph, Frosty, etc. My grandmother always signed hers as Santa and it wasn't a big deal because she bought our Christmas outfits as our gift so there were no questions.
We haven't really decided what we are going to do for DS. I think DH and I put our names, or goofy characters, as the from and not Santa but I really don't remember.