He sounds like a twat OP.
There's loads of other stuff in your post which are equally arsehole behaviour aside from his tantrum over the car being washed.
For example, not only does he get to enjoy a few drinks and be driven there and back, you also entertained the kids. He is responsible for washing the car, but didn't so you've had to do it yourself. Doesn't sound like he was grateful for any of those tasks.
If he is particular about his car then he needs to do it himself. I'm not sure he has a right to be particular if it's a family car that you're both using, and certainly not when he has judgemental parents.
Honestly OP you sound like a wonderful, thoughtful person, and I wonder if you might be better off without this ungrateful man and his judgemental parents.
Did he think you were going to wash it by hand while also solo parenting (sorry don't know kids ages)?
If the kids are teenagers then he could have offered them some pocket money to clean it if he wants it done by hand with a specific shampoo.
He's been absolutely ridiculous. One go in the car wash isn't going to ruin the paintwork. And if he's so worried then he should do it himself.