If I bought a present for a child and gave it to the parents before the child's birthday because my family and I were going to be out of town, then later the parents have a birthday party, do I get the child another gift?
Only if you want to. Both the parents and the child will know the gift has been given. The only difference between giving it before and during the party is who knows. If in your area of the planet everyone knowing is important, then make it known; otherwise, it is already known by those who matter, so party on!
No, the family knows that you have already given the child a gift. If you feel more comfortable showing up with something for your own satisfaction, or so others won't think you showed up/didn't give Jr. a gift at all, there are a few options. Take a card, slip $5 in it if you'd like, get a small/inexpensive accessory or coordinating piece to be used with the original gift: say you bought a racetrack for those little cars (sorry, I have a girl!)....get a couple more little cars to go with...bought a DVD of Spider-Man..get a little SM action figure or cool SM reusable tumbler. That way, should they open the gifts at the party, you can just say something like 'I wanted to get you something to use with the 'real' (laugh) gift I gave you a few weeks ago' or 'I'm so happy to hear you're enjoying the Batman car I gave you a bit early, so when I saw this 'xyz', I couldn't resist adding to my original gift.‘ On the other hand, nobody should fault you for not bringing another gift, and probably won't even notice.