The way this question is phrased already assumes the answer ‘no’ – that we should not – because indoctrination is a negative word today. One young adult objected to teaching our children biblical perspectives because this was indoctrination. Rather we should ask, should we disciple our children? And of course, my answer is certainly, for Deuteronomy 6:4 ff. commands us to teach the words of God diligently to our children. We are to talk of them when we walk by the way and when we lie down and rise up.
I am the product of a Jewish and Norwegian marriage. My mother and father agreed to not teach us either in the religion of Judaism or Christianity but to allow us to choose our own path when we got older. This seemed very enlightened but was mistaken. Nevertheless, God called me to Himself when I was twelve and a half years old. When we do not disciple our children, we leave a vacuum. They will then be indoctrinated by the larger culture around them, the schools, the internet, and social media. We desire that parents would be in control of those sources of indoctrination.
However, it is not as if we desire to produce young adults who cannot think for themselves in dependence on the Spirit. When children are young, they learn basic biblical truths, the memorization of Scripture, Bible stories, and great stories of those in missions. As they enter young adulthood, we need to deal with difficult questions: the problem of evil and suffering, other religions, the secular world, the nature of justice, and even economics. If parents are not capable of such a dialogue, they need their young people to be connected to those who can deal with such issues. In addition, they need to be able to understand both sides of an issue, even the arguments of those who are against biblical faith. The young adult then learns to engage in such a way that they are able both to see the other side and not to be taken in by it. They can weigh arguments and are not merely indoctrinated. However, walking in the Holy Spirit will be a key to not going too fast beyond the capability of the young person to absorb and deal with the issues. Seeing mature, discipled and educated young adults should be our goal.