Re: What to do :
I wouldn’t do anything. This is not a big deal. There are other major issues to worry about in life than whether or not someone said “Eid Mubarak” to your parents. Especially since you two are not married and he doesn’t live in the same city..so I’m assuming that you and your parents did not see him on Eid day.
A lot of men (My SO is one of them) aren’t good at remembering special occassions and calling on that day. My SO forgot to call and with his brother a happy birthday this year! SO’s dad last year had to call him and remind him to call and wish his mother a happy bday! My SO doesn’t do it to hurt anyone or b/c he doesn’t love his parents…it just slips his mind.
Once you get married and are living together, this wishing “Eid Mubarak” to your parents won’t be an issue…so right now this isn’t a big deal.