My daughter came home from school and reported that a classmate claimed her parents do not allow her to have birthday parties because they were "invented" by Pharoh. Is there any truth to this? Is there a prohibition against birthday parties?

It is true that the first and only birthday party mentioned in the Torah is Pharoh's. However, that doesn't make it prohibited. Rav Ovadiah Yosef (Yabia Omer Vol 6 OH 29) mentions several opinions that approved of birthday parties, especially if they include words of Torah and praise for Hashem. Rav Moshe Feinstein also mentions having a birthday party at home without disapproval. The Chatam Sofer felt it was improper to have a birthday party, but approved of a yearly celebration of the day of circumcision for a boy. (I am unaware if he talks about a girl.) I am aware of Poskim dissaproving of parties that spend a lot of money and have no Torah content. I am not aware of a blanket prohibition.