Having been a engineering manager before, it would put an extremely bad taste in my mouth if someone waited until the offer stage to ask for a part time role.
I can understand why you feel that way. But many companies simply doesn’t want to talk to people who wants to work part-time. (As in ”don’t call us, we’ll call you”.) So even if it might seem pushy to negotiate like that, at least to me it’s understandable.
It also has the potential to waste your time and theirs. I wouldn't call it pushy as much as borderline deceptive.
Unless the employer posts the salary up front, they are bring deceptive too.
While that is valid, two wrongs don't make a right.