You're the one with the strength in this area (finding child care), and I think you should do as much as you can. I get that you have already done a lot, and you feel that he has done nothing.
But this *is* your strength, and you should want to help him. And to help your family.
If he's been treating you in a way that makes you not WANT to help him, that's a little bit different conversation.
Have you two had a really open conversation about this?
Time for one!
Be absolutely certain that this conference is an opportunity that he CAN pass up, and not something that will really hurt him.
And then paint the parallels, perhaps.
He wants to go on a conference by himself; you want to go on a trip with your parents by yourself.
It didn't sound like there's any request for you to cancel--just badgering for you to make something happen that isn't happening.
So maybe in that conversation, you need to clearly delineate all the routes YOU have tried, so that he sees, in one clear place, all you have done.
And then say to him, very clearly, "I've done all I can. Now your reminders have crossed a line into badgering me. And the sad truth is that it is making me pissed of at you and making me resent you. I don't want that dynamic between us."
And then give him guidance on what more he could do himself to line up child care. Say to him very clearly: "I want to make this perfectly clear, hon. Now YOU have to do the work."
Then, when he mentions it, you can say, "What did you do about it? Have you called anyone? Have you taken any of those steps?"