I'm not a big believer in "the whole class" idea. I think it puts a lot of pressure on parents to go to every party, which becomes expensive. I think a lot of kids get the invitations and say "Who?" I think 3 year olds can't sit around and watch one child open 13 gifts and not wonder why they can't have them. I think overloading a birthday child with 10 or 13 (or worse, 25, when they get to elementary school) gifts just teaches them to be greedy. And if they don't open the gifts in front of their friends, the friends learn that a gift is just the price of admission to a pizza or fun space party.
Then the parents spend all year trying to afford gifts for one party after another. There's the guilt of not going to a party of someone who came to yours, and the pressure to take your child to a party being held at a "cool" party place, even if it's not a very good friend. You see tons of questions on Mamapedia about people trying to find an affordable venue for a whole pile of kids, then worry about other parents bringing siblings who expect pizza and a goody bag, and it just mushrooms.
It's much easier just to keep it small. When our son was small, we used the "age rule": Shen you're 3, you invite 3 kids. When you're 6, you invite 6. By the time they're 10, they just want to take a few kids to the movies or have a sleepover rather than a conventional party. So it shrinks down on its own.
So no, don't invite everyone. But that also means not to give out invitations to some families in front of the others. And don't expect the day care center or preschool staff to give out the invitations. You need to invite people you and your child know well enough to have a home address or at least an email address.