I'll get a lot of thumbs down for my answer but well be warned, I don't care.
A birthday, like a funeral is a celebration of life. Myself, I would have still had the party for my kid(s).
I was a young adult when my dad died, but lost a parent young none the less. He was sick for 10 years so his death was expected. The day of my dad's funeral, after it my mom and I went to a candle party of some close family friends that we were planning to go to before he died after that we went bowling with the same family friends and my brothers joined us. This was actually a bowling league and we missed a night in the few days after he died so we were making up our games. These family friends had lost their dad about 6 months before my dad died so these were actually people who understood what we were feeling.
My brothers birthday was the day after my dad's funeral, later in the week we still did something for his birthday.
Celebrating something good or doing something fun is a whole lot better than sitting around and feeling sorry for yourself. Me sitting around my house feeling sorry for myself wasn't going to make my dad less dead and keeping myself occupied also didn't make me grieve less, it helped us start to move forward.
I also wouldn't want my family to sit around and feel sorry for their selves. I let it be known when I die, just have a party or otherwise a celebration not a funeral or go on a trip and grieve me in the Bahamas or anywhere else they'd like to go.