Birthday should be celebrated on a more convenient date, like the weekend before/after.
This is the norm for my extended family (and a large portion of our friends), as almost everyone works in a field where the work has to be covered 24/7 no matter what day of the year it is: many of us have limited options on when we can book leave, rotating days off instead of fixed weekends, and morning/afters/nights shift patterns to consider too (forces/fire-brigade/police/nurses/carers/farmers/etc).
And its not just birthdays that are moved - I don't think I've celebrated Christmas on the 25th of December more than twice in my life - one year it was February before we managed coordinating days off, and another we shifted it to April Fools for the fun of it
!
We see it as being extra special because it's moved, not less special - we have our own Christmas, and because it's not the day that all the other sheep people celebrate it, we can have our friends over too (which is a big hit with the young people
), we don't have the mad rush on at the supermarket because it's Christmas eve, and it's usually cheaper too
.
It's the same with birthdays, want a summer BBQ/pool party when your birthday is in the middle of winter? - celebrate it 6 months later. Birthday falls on Christmas/Valentines or is the anniversary of when a loved one passed away? - pick another date, celebrate then.
If it's really that important to the parent staying home that they see birthday-child within an arbitrary fixed 24hr period, then they can either video chat (every mobile I've had in the last 10 years has had the ability), or hire a car and drive there straight from work to holiday destination (or meet up part way at a restaurant if that's too far).
I suspect parent staying at home, consciously or not, feels left out, and thinks it's a bit unfair that everyone else gets to go away without them, having fun while they have to stay and go to work.... and perhaps needs some reassurance that they are wanted/needed/loved/will be missed...
I've seen it before when one parent has worked away (eg military) where it can lead to feelings of insecurity along the lines of 'they're managing just fine without me, they don't need me, they didn't even miss me, they don't love me any more...'
So perhaps?
weekend before: child birthday,
mon-fri: holiday away,
weekend after: something special/extra attention for parent who couldn't join?