"Eve, the wife of Adam, became the "mother of all the living," as we read in Genesis, chapter 3, verse 20. Because of Adam and Eve's disobedience to God, sin and death entered the world, and all of Eve's children are born in original sin. Through the obedience of Mary, the handmaid of the Lord, Jesus Christ entered the world as man to reverse the disobedience of Adam and Eve. For this reason, Christ is considered the "New Adam." Christ, the Son of Mary, offers a new life of grace to all who believe in Him. All who accept Christ's invitation become His brothers and sisters, and become the spiritual children of Mary, the New Eve."
The Scripture places the blame for sin entering into this world on Adam rather than Eve (Romans 5:12-21). The reason for this is that he received firsthand knowledge from God and still disobeyed. Paul develops a typological parallel between Christ and Adam: the former brought life upon mankind whereas the latter brought about death (1 Corinthians 15:45-47). Therefore, any notion of Mary being the New Eve collapses because the responsibility for the fall is attributed exclusively to Adam (even though the woman had sinned). The Apostle Paul does not extend this parallel to Eve.
Early Christian writers such as Justin Martyr made parallels between Eve and Mary. Irenaeus wrote the following in his Against Heresies:
"For just as the former was led astray by the word of an angel, so that she fled from God when she had transgressed His word; so did the latter, by an angelic communication, receive the glad tidings that she should sustain (portaret) God, being obedient to His word. And if the former did disobey God, yet the latter was persuaded to be obedient to God, in order that the Virgin Mary might become the patroness (advocata) of the virgin Eve. And thus, as the human race fell into bondage to death by means of a virgin, so is it rescued by a virgin; virginal disobedience having been balanced in the opposite scale by virginal obedience. For in the same way the sin of the first created man (protoplasti) receives amendment by the correction of the First-begotten, and the coming of the serpent is conquered by the harmlessness of the dove, those bonds being unloosed by which we had been fast bound to death."
The articulated parallel does not amount to the full theological developments within Roman Catholicism. It is not about Mary being a mediator of grace or an intercessor of sorts. Simply put, parallels are established between the fall and redemption with Jesus Christ being the central figure of preeminence as the new Adam.
Mary is like Eve in that she is a woman and that she gives birth to the promised seed. However, any idea of there being a salvific parallel between Mary and Eve goes beyond what is taught in Scripture. The authors of the New Testament nowhere apply a concept of divine motherhood to Mary over the church. Mary is blessed in that she found favor with God, but she is not greater in holiness and grace than any other believer in Jesus Christ. Mary is not the mother of a new humanity any more than was her own mother.
The only woman mentioned in the New Testament and given the title of "mother" in a spiritual sense is Sarah (1 Peter 3:6). However, even that status is due to the fact of her being a good moral example rather than anything relating to saintly intercession.