Let me first quote the Catechism (my emphasis in red, with special emphasis in bold red):
The consequences of Adam's sin for humanity
402 All men are implicated in Adam's sin, as St. Paul affirms: "By one man's disobedience many (that is, all men) were made sinners": "sin came into the world through one man and death through sin, and so death spread to all men because all men sinned." The Apostle contrasts the universality of sin and death with the universality of salvation in Christ. "Then as one man's trespass led to condemnation for all men, so one man's act of righteousness leads to acquittal and life for all men."
403 Following St. Paul, the Church has always taught that the overwhelming misery which oppresses men and their inclination towards evil and death cannot be understood apart from their connection with Adam's sin and the fact that he has transmitted to us a sin with which we are all born afflicted, a sin which is the "death of the soul". Because of this certainty of faith, the Church baptizes for the remission of sins even tiny infants who have not committed personal sin.
404 How did the sin of Adam become the sin of all his descendants? The whole human race is in Adam "as one body of one man". By this "unity of the human race" all men are implicated in Adam's sin, as all are implicated in Christ's justice. Still, the transmission of original sin is a mystery that we cannot fully understand. But we do know by Revelation that Adam had received original holiness and justice not for himself alone, but for all human nature. By yielding to the tempter, Adam and Eve committed a personal sin, but this sin affected the human nature that they would then transmit in a fallen state. It is a sin which will be transmitted by propagation to all mankind, that is, by the transmission of a human nature deprived of original holiness and justice. And that is why original sin is called "sin" only in an analogical sense: it is a sin "contracted" and not "committed" - a state and not an act.
405 Although it is proper to each individual, original sin does not have the character of a personal fault in any of Adam's descendants. It is a deprivation of original holiness and justice, but human nature has not been totally corrupted: it is wounded in the natural powers proper to it, subject to ignorance, suffering and the dominion of death, and inclined to sin - an inclination to evil that is called "concupiscence". Baptism, by imparting the life of Christ's grace, erases original sin and turns a man back towards God, but the consequences for nature, weakened and inclined to evil, persist in man and summon him to spiritual battle.
408 The consequences of original sin and of all men's personal sins put the world as a whole in the sinful condition aptly described in St. John's expression, "the sin of the world". This expression can also refer to the negative influence exerted on people by communal situations and social structures that are the fruit of men's sins.
In CCC 405, you must read the whole sentence. While the sin is remitted by Baptism the consequences of sin, and sin, remain. The Church states this, "...but the consequences for nature, weakened and inclined to evil, persist in man and summon him to spiritual battle."
In response to your direct question, Original Sin is imparted to each individual "Adam's sin and the fact that he has transmitted to us a sin with which we are all born afflicted" (CCC 403). Thus, each person must be baptized.
If the parents' baptism wiped out original sin for the children, why then baptize the child in the first place? This would led to the heretical notion of a "believer's baptism" in which it is thought that babies do not need baptism. That is a notion of many Protestant sects (e.g., Baptist, Fundamentalist, Evangelicals, etc). No, original sin is transmitted to each person, independent of parents baptism, and thus each person must be baptized to remit original sin (and personal sin in the case of an adult baptism)..
I hope that helps.
Bro. Ignatius Mary