A parent's perfect friend is infinitely patient - like the perfect parent, they will put up with hours of crying, babbling nonesense, capriciousness, inconsistency and obstinance - all the while cheering for more... and they'll help with the child too.
A parent's perfect friend has a maginificiently exciting life with which to regale the bored parent, yet be available whenever the parent needs them, and never tired of hearing stories about the child's sleeping / eating / pooping schedule or lack thereof.
A parent's perfect friend will be independently wealthy, and thus available during weekdays and at 3am and able to treat the parent to lunch... somewhere with beautiful views and delicious food and safe play areas.
A parent's perfect friend will be seriously interested in spending stacks of time with a strung-out parent and their pre-verbal child, yet - of course - have no children of their own.
A parent's perfect friend is - clearly - imaginary. :)