I was just reading a FABULOUS book called "That Was The Best Party Ever" by Sharron Werlin Krull, and she has an excellent way of handling the present opening portion of any child's party! Often, this is where the party dissolves into chaos... all the kids are shoving their gift into the birthday child's face, others are begging to play with some of the toys that have been opened... and it's easy to see why some parents try to avoid it all by opening the gifts after the guests have left. However, opening gifts later doesn't allow the guests to enjoy seeing the birthday child's reaction and how their gift was appreciated! And it doesn't allow the birthday child to look them in the face and say "thank you!"
Sharron recommends solving all these problems by turning the gift opening into an organized game, in fact the last game of the party. The basic game is that all the kids sit in a circle, with their gift to the birthday child behind their back. The birthday child sits in the middle of the circle. An adult or older child has charge of a large trash can or trash bag for wrappings, and another adult controls the music and takes notes on who gave what (to help with writing thank-you notes later!). A large stuffed animal or other item (in keeping with whatever the party theme was! So for example, perhaps a wand at a fairy party, a bear dressed as a pirate for a pirate party, and so on) is passed around the circle as music is played. When the music is paused, whoever is holding the item gets to give their gift to the birthday child! The birthday child opens the gift, and stands up and turns around so everyone can see the gift and "ooh and ahh" over it, then the birthday child turns to the child who gave the gift (it's easy to remember who, because they are still holding the pirate bear or whatever!) and says "Thank you _________!" Then the gift is given to an adult who puts it away for now. No opening packages or taking toys out, because they are in the middle of the gift opening game! Then, the music starts again, the bear or other item is passed, and the game continues until everyone has had a chance to give their gift to the birthday child!
This is a perfect activity for while the parents are arriving to pick up their kids, as they will doubtless be impressed by how calm and well-organized all the kids are! Plus, the adult in charge of the music can subtly control who gets to give their gift next, so if someone needs to leave before the game is finished, that can be easily arranged. :-)
There are a lot of other variations to this game too, such as:
My adventures as a party planner and kids' entertainer! And lots of party inspiration too!