A caesarean section [with epidural anesthesia and ardent support in the form of his mother] is not the worst or even unpleasant thing Percy has experienced.
Most of all, Percy is afraid for his daughter - the doctors said that she was a big really big little girl, Jason joked that she would look like him, or some even assumed that there were more than one child, although this assumption was not viable - there was only one confident and strong heartbeat.
He was afraid that there might be something wrong with her, that she might be sick or... something he didn't want to think about.
But everything seemed to be going well, - there was a cutting slightly irritating sensation, the soothing voice of his mother and the reassuringly confident behavior of the doctors, until his girl left the cradle of his body and cried, the doctors and Sally saw her and... they were surprised and shocked.
Percy, having experienced fright, demanded to immediately show her to him and they obeyed him, the girl was presented to the level of his face so that he could easily see her. At first he thought she was beautiful and perfect with that delicate skin, sleepy tear-stained sea-green eyes and light fuzz on her head, until he noticed something big surrounding her on both sides... The large wings, softly and neatly covered with downy plumage, are dull gray like those of an eaglet. Because of them, they couldn't even swaddle her and just covered her with a blanket.
eagles were the sacred birds of Zeus/Jupiter, Jason the father of their daughter was the son of Jupiter...
"damn it," Percy thought, "I just gave birth to an eaglet... my daughter would be stuck in my pelvis if it were a natural birth".
He did have a few questions for Jason and perhaps the king of the gods himself.
Little Rhea Ariana eventually got the nickname Aëttie - from the ancient Greek eaglet.
I could just imagine little Aettie wadling up to her parents asking them why all of the other kids don't have wings like her. And both of their parents having trouble answering because they don't know either, even with the godly parent lineage. Eventually they just end telling her that she's just a very special girl.