I posted on this thread the way to work, and now I've finished for the day I've read the replies.
First - OP is being a bit antsy to posters and for that reason I'm a bit meh.
But the principle of the discussion/request is a good one. I found it very hard when my DC were small because i often had to work late (most days) and i did have some inter(continental)national travel. My DH worked split shifts in a restaurant which made morning drop offs ok, but everything else was between me and the childminder (for whom i paid an absolute fortune. I considered it, for the years it was hideously expensive, an investment in my career and our family future.
I sharp elbowed my way up from assistant to senior manager at that company, and the early work i put in, my reliability and willingness to put in the hours/miles/work paid off. The other people with children were all men, and their wives mostly worked around their children, if they worked. That was their family decision (although over the years more than one complained about their lack of career to me, and were "jealous" that i was independent and senior - compared to their jobs.)
What i think? it would be a good recruiting tool if companies would contribute to extra out of pocket expenses like this, on a case by case basis with no assumption that it will always be extended to all employees. Where there is a parent at home? sure, they need to coordinate. Is the employee (OP in this case) really needed on this trip? If not: can they impose this trip on someone else (and not always the childfree/childless and those with no dependants. Also they could offer it to employees with other caring responsibilities. If the company considers the employee valuable, they will offer more inducements/perks.
From OPs POV, she said she's not needed, it won't increase her already negligent promotion prospects and she doesn't want to go, and is unwilling to ask her DH to step up. So there are 2 choices: ask your company to contribute/cover the extra costs (and suggest they review their policies in general about this) or decline the trip.
The reason, i think, some pp say this is why other people don't like working with parents is that they often get landed picking up the slack when little Johhny/Joanna is sick, is in a play, etc. They might want holidays in times when all the parents have booked so they can't. They might have to cover Christmas/Easter because "they have no families". The more men step up and also ask for this perk (to cover extra childcare expenses) the better.
But in this day and age? you will have to be a hugely desirable employee for a private company to stump up this kind of extra support. I'm interested how it ends with OP