Santa brings a stocking for everybody (mummy and daddy included, and any guests staying - albeit the grown up ones are much smaller and usually just have the satsuma, chocolate coins, choc brussels sprouts (ongoing family joke around these!), bar of luxury choc and a couple of poundland or ebay bargain gadgety bits).
Used to have santa wrapping paper that was different for both dc but now just have one santa paper (which I have to remember to hide well away both before and after christmas!) and for the stocking presents, a couple of years ago I bought a stash of cheap stripey/dotty green/red sweety bags in a couple of sizes to wrap the presents in instead. It's great - no faffing around with sellotape, you can just twist and scrunch the presents into the bags and they stay wrapped, dc haven't noticed and still need to tear off the paper as if wrapped normally. Takes significantly less time to wrap everything and hopefully it means that the paper recycles better due to the lack of sellotape. Occasionally there is a slightly bigger parcel that doesn't fit into a single bag but so far I have been able to wrap it suitably using two bags - although I might use a dab of sellotape on these to keep them together.
these are the sort of bags I've used (although it's worth doing a hunt on ebay and amazon if you do want to use them - prices seem to vary a lot depending on how many you want to buy at a time and colours etc - chances are you'll get a cheaper set than these with a little look but these are still reasonable and have plenty of colours)
he also brings a single present downstairs under the tree for each dc that's in his own paper - not the dc's main present but a reasonably nice one (one year they got a lego baseplate each and a big box of sharing lego, one year we got them a games console and they got a game each but a single controller to share (the games console came with one, wanted them to have one each, thought it easier that neither had claim to a particular controller!), and so on.