I'm comfortable with dropping the indefinite article here, and I think I may start to be having a better intuitive grasp on when to drop it. However, this article doesn't seem to state a rule that applies specifically to this sentence. There may be a rule, but it's not listed in this article. Is this sentence about dropping the indefinite article when indicating possession?
Well, one may have a party in honor of their birthday but it may take place on a different day, say the weekend. This sentence specifically says that the person is having a party ON their birthday. So, while the two sentences may overlap in usage, they technically do not mean the same. Hope that is helpful