Let's bracket all moral concerns, and simply suppose that the parent does not count sexual activity by her minor children as having positive utility (or at least counts it as of such low utility as to be negligible), but does count a pregnancy (in the child or caused by the child) as a significantly negative outcome. Rhetoric from those advocating greater availability of contraception to children suggests that such a parent would be instrumentally irrational to object to the child's receiving contraception.
However, this is mistaken. Typically, children also consider a pregnancy a significantly negative outcome. Therefore, in a large enough population, there will be children who would be very unlikely to engage in sexual activity if there is a significant danger of pregnancy, but if that danger were significantly reduced, would engage in sexual activity. In the case of such children, it may very well be the case that the availability of contraception increases the risk of pregnancy. For instance, suppose that without contraception, over the period of a year the child would have a probability of 0.98 of not engaging in sexual relations at all. But if the pill is made available, the child has a probability of 0.50 of using it and having an average sexual frequency for sexually active persons.
Now, if contraception is not made available, the likelihood of a pregnancy is (0.02)(0.85)=0.017, where I shall suppose 0.85 is the probability of conception without contraceptives at an average sexual frequency for a sexually active person. This is actually an overestimate of the likelihood of a pregnancy, since if the child is afraid of a pregnancy outcome, the frequency is likely to be significantly lower. If the pill is made available, the likelihood of a pregnancy then will be (0.50)(0.05)=0.025, where the 0.05 is permthe typical-use failure rate for oral contraceptives.
Therefore, a parent who knows with a sufficiently high probability that her child satisfies the above assumptions and seeks to prevent the child's pregnancy will be instrumentally rational in refusing contraception for that child.
Moreover, since there surely are such children in the population (there is, obviously, a broad distribution in the attribute of caution in teenagers, and there are teenagers who are very cautious), it follows that even if making contraception available to all teenagers were to reduce the overall pregnancy rate (and I am not aware of any data that it would), there would be some individuals the risks for whom would be increased by the availability of contraception. And, of course, there will be individuals the risks for whom would be decreased by the availability of contraception—namely, those who would have a sufficiently large sexual frequency even without contraception. Therefore, making contraception available to all teenagers results in a redistribution of risks—some come to be better off pregnancy-wise and some come to be worse off.
Now, while it can be licit to have a public health initiative that redistributes risks, increasing those of some and decreasing those of others, significant gathering of empirical data is needed before any such policy is put into place, to ensure not only that the overall risk is decreased, but also that no subgroup's risk is increased in a way that is morally unacceptable. For instance, if a chemical added to the water were to improve the dental health of a majority ethnic group but decrease the dental health of a minority ethnic group, the introduction of that chemical would be morally problematic—significant amount of information-gathering would need to be done, and attempts to limit the application of the initiative to the minority might well need to be made (e.g., not adding the chemical in the areas where members of the minority group are more likely to be found).
In particular, the following empirical outcome is imaginable. It could be that the availability of contraceptives significantly increases the likelihood of pregnancy among religiously conservative teenagers, because without the availability of contraceptives they have two reasons to avoid sex: (a) religion and (b) pregnancy (and disease, but what I say about pregnancy applies to STIs mutatis mutandis), which two reasons may result in a high probability of abstinence and hence a close to zero pregnancy rate (rape can happen despite abstinence, so it's not exactly zero), while with the availability of contraceptives the second reason largely drops out, and the abstinence rate may significantly decrease. If that were so, then there would be an identifiable group for whom the risk of pregnancy would be increased by the availability of contraception. I do not know if it is so or not—that is an empirical question, and either answer is possible depending on how the probabilities work out. But it is not irrational for parents of religiously conservative children to worry that the availability of contraception might increase the risks of pregnancy for these children, and it might well be irrational to be confident that it does not increase these risks unless one has significant empirical data (of which I am not aware).