All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
We do not have any sound evidence in the Sharee'ah that proves that the recitation of Chapter Maryam or Chapter Yousuf helps ease labour pain. No doubt, believing so or practicing it is an innovation. The Prophet said: "Whoever introduces in this matter (religion) of ours something which is not part of it, it is to be rejected." [Muslim]
It should be known that Allaah made the whole Quran a cure; as Allaah Says (what means): {And We send down of the Quran, that which is healing and mercy for the believers.}[Quran 17:82]
However, singling out the recitation of a portion of the Quran for a given purpose without a sound evidence is a form of innovation because it is specifying something which the Sharee'ah did not specify.
However, a woman can ask Allaah to ease her pains and make her delivery easy or whatever else she wants.
Allaah Knows best.