Parto (Spanish for “birth”) comes from the Latin partus, “brought forth”. That makes sense: a baby is just brought forth into the world.
From the same Latin root, we get the English partum for “birth”. But that word is really only used in one contemporary word today: post-partum depression, the depression a woman gets after childbirth. Yes, post-partum is merely “after-birth”.
The p‑r-t root is clearly visible in both words.