I think a birthday in Paris would be delightful, and you're lucky to have a family that's thoughtful enough to take you there!
Having said that, August is the hottest month in Paris, and it can be humid. Additionally, many Parisians head off on holiday during August, and there is a huge influx of tourists.
Are these all necessarily bad things? It depends on your perspective. If it's hot and humid, there are many things to do indoors under airconditioning -- the Louvre and Les Invalides, to name just two, are "must sees" anyway. Will you miss the native Parisians? Probably not, assuming you aren't French. And as for all those tourists . . . well, YOU will be a tourist, won't you?
If someone offers you a trip to Paris as a gift to celebrate your birthday, my recommendation is that you pounce on it. You don't get enough birthdays in life, and this will be one you'll always remember!
Have a great trip!