After parcel post service was introduced in 1913, at least two children were sent by the service. With stamps attached to their clothing, the children rode with railway and city carriers to their destination. The Postmaster General quickly issued a regulation forbidding the sending of children in the mail after hearing of those examples.
That photo is part of the Smithsonian Institution’s collection at Flickr.
Update: A 1913 NY Times article includes a query from a citizen to the Post Office inquiring whether they could send a baby through the mail:
Sir: I have been corresponding with a party in Pa about getting a baby to rais (our home being without One.) May I ask you what specifications to use in wrapping so it (baby) would comply with regulations and be allowed shipment by parcel post as the express co are to rough in handling