It's rude.
I have similar issues with standing on benches with fabric cushions and kids being sat on TOP of my cubby units. I have a laminated sign that says "Please do NOT place your child on top the cubbys. Thank you!"
Most parents can't read it though because their child's butt is blocking it from view.
In your case, I'd have a hard time not putting something on the table that will spill over and make a huge racket when DCD puts the coat on it. Something like a tall container full of little bells or acorns or Christmas tree bulbs etc... something so that the minute DCD sets the coat on the table it falls all over the place.
Then I'd stand there and say
"That's why I've asked you not to put her coat there Bob. Could you please make sure that is picked up before the another child arrives? Thank you"
I do agree.... people can be obliviously rude sometimes.
