This is not a question about the literal translation of the sentence: "I'm proud that you are my dad." The question itself is more like: what do Chinese people say when they want to express their feelings to someone that they are proud of? Like, when children are proud of their dad?
The simplest sentence would be something like:
So in your case：我为爸爸骄傲。
Slightly more sophisticated:
@Ha Hae's 我以你为荣 would also work but would translate to something like “You are in my place of honour."
I've recently watched the movie "A Love So Beautiful" (致我们单纯的小美好) and learned the sentence "我以你为荣", meaning I'm proud of you. It's right on the first episode. By the way, this movie is so lovely and you can learn a bunch of interesting phrases for sure, I highly recommend to everyone ;)