My budgies? They poop wherever they want to whenever they want to.
I noticed Pipp sometimes tries to hold it in a bit if she's on my shoulder without the towel, and if the towel is on she poops on it. When hold her, as soon as I put the towel on my shoulder she hops right on! She knows the towel is there for a reason and she should not jump on unless I put her there or the towel is there. Smart bird, huh? I didn't even train her to do it!