genitals

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Yes and no. Both men and women have genitals (L. genitalia, also plural).  They are complex organs, partly internal and partly external.  Hence, the plural refers to their having several parts.  By usage, however, it is a plural noun referring to the entire apparatus, so one part is not called a genital-- this latter is an adjective only.

Your genitals are your private parts. (not 'part')

How often does it come up in casual conversation??

Hi New2grammar

Although you've used 'part' all along, don't worry, your parts are still intact.