@CorkyBebe, I think you might be out of time to host a birthday party for that particular sim. A sim can host a birthday party if you still have the space to add a new trait for them, but based on your photos that sim already has four/five traits and is at the retirement stage, so based on what others have already posted, it's too late for them to have birthday parties. They can host other types of parties still, though!
If I've passed up forever the chance to host a birthday party, how on earth can I ever host a party of any description?
@Ivanshoe have you tried my suggestion? https://answers.ea.com/t5/Game-Information/can-t-host-party/m-p/7814201/highlight/true#M20518
