I googled "Welcome to the Birth Pangs Birthday Party!" and found the whole sentence from a blog named "Birth Pangs", so it is quite clear that they were just celebrating the birthday of that blog.

Birth pangs = the pains of the mother just before and when giving birth "Birthday pangs" is undefined in the dictionary. I guess it refers to nervousness that one has toward an upcoming birthday celebration.

Pangs is a popular person. You can put a person's name at the end of the phrase "happy birthday' Smile!"

I'll have to believe Jesse, to me it means emotional pain. People don't like getting older, so maybe someone took the sadness from getting older and turned it into a bit of a joke!? Birthday Pangs Party is something I have NEVER heard of - must be American, or something kids are saying. Sorry!