My parents are both Christians. When we were kids they always took us to church. Does that make me a Christian?
God knows each one of us as a person, not just part of a family. This is clear even in the Old Testament:
Ezekiel 33:20 (ESV)
Yet you say, 'The way of the Lord is not just.' O house of Israel, I will judge each of you according to his ways."
Looking at the Gospel in the New Testament, we see that again each of us is an individual to God:
John 3:16 (ESV)
"For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.
God saves whoever believes in Jesus from perishing. Only by believing, yourself, can you be saved.
It depends what you mean by "Christian". The term has been used for millenia to refer to Christ's followers. However, one can follow Christ for a time, but never have truly believed in him, and eventually fall away. On the other hand, I typically associate "Christian" with someone who is truly saved. The term can mean different things to different people.
In fact, there have been some terms coined just for these types of discussions. One such distinction people have made is between the "invisible church" and the "visible church". The visible church is made up of, essentially, people who self-identify as Christians. Those whose parents are Christians and take them to church every Sunday would fall into this category.
I would even go further to argue that there are certain privileges afforded children of believing parents. These children may have the opportunity to hear the gospel and scripture on a regular basis. The may see modeled how a Christian marriage and family should look. The may learn how to pray. They may have the loving support of a church body.
However, these privileges are earthly privileges.
On the other hand, is the invisible church. This is made up of all people who truly belong to the Lord, both living and dead, whether or not their parents are Christians, and whether or not they were in church this past Sunday. The privileges afforded these people is beyond compare.
One thing you will find made clear in the New Testament time and time again is that salvation is through faith alone in Christ alone.
In Jesus' day, it was thought that being a descendent of Jacob was sufficient for salvation. Paul rejects this explicitly at several points in the book of Romans. In Romans 2-3, he argues that Jews are not automatically saved, and again in Romans 9, he says "not all who are descended from Israel belong to Israel" and "This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring." (ESV). It is clear, then, that it is your own faith, not that of your parents, that causes you to be numbered among the saints.
We see that John the Baptist was a Christian from the time of his birth.
Luke 1:66 (NKJV)
66 And all those who heard them kept them in their hearts, saying, “What kind of child will this be?” And the hand of the Lord was with him.
So it IS possible. But in reference to your question, your parents are Christian, so what does that make you? The answer is:
1 Corinthians 7:14
For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband; otherwise your children would be unclean, but now they are holy.
A Christians children are Holy because the parents have been blessed as a believers. Does this mean that you have already been saved from God's judgement? Unfortunately not. Because we know that each person will be judged by their own actions and not the actions of others.
Catholic teaching is that you must be born again to be Christian, "unless one is born of water and the Spirit, he cannot enter the kingdom of God" (John 3:3-5). This second birth is the sacrament of baptism, where we are joined to Christ as members of his body (1 Corinthians 12:13, Galatians 3:27).
Baptism can be given to newborns, older children or adults. Baptism of little children is based on the faith of their parents, who promise to raise them in the faith. Baptism of older children and adults is based on their own profession of faith.
It is possible for those who have become Christian through baptism to later leave the faith by their own choice, thus becoming apostate, so baptism is no guarantee of salvation.