Torture to start with, private parts getting rejected at a party (9)
I believe the answer is:
strappado
'torture' is the definition.
(thesaurus)
'start with private parts getting rejected at a party' is the wordplay.
'start with' indicates taking the first letters.
'getting rejected' is a reversal indicator.
'at' means one lot of letters go next to another.
'party' becomes 'do' (do is a kind of party**).
The first letter of 'private' is 'p'.
'p'+'parts'='pparts'
'pparts' back-to-front is 'strapp'.
'strapp'+'a'+'do'='STRAPPADO'
'to' acts as a link.
(Other definitions for strappado that I've seen before include "be nasty - with this" , "Form of punishment" , "form of torture" , "method of torture" .)