Actually in English body parts and parts of the body are often used in slightly different contexts. Certainly what the police find in crime dramas are body parts, but in anatomy class you are more likely to talk about the parts of the body. Both are correct in English, but they both have the same Spanish translation.
If you are talking about The parts of the body you are talking about it all. It isn't generally a euphomism, although it could be. But one of the reasons I learn the vocabulary forthe parts of the body when I learn a new language is to be able to talk to a doctor. So I don't only need to talk about my arms, legs, hair and face, but also my heart, kidneys, and yes, my "lady parts". So I am looking for the more clinical words, not the crude words.