I had the same problem with my playgym its coloured and when you go to clean it the colour run's I just took the whole gym ran it under super hot water till it stopped bleeding and that seemed to work for me
lol poor pink baby. i can't see how i wouldn't come off it washed it but yet come off on the birds. The dye itself shouldn't hurt them. Yes it will turn them a different color and they make look silly.
I've had a dyed play stand for ages. i've only ever seen the color run when it got wet. My budgies are always rubbing their heads over it.