The Vatican does not like assisted procreation, this is well known. The arguments in support to that are, however, very weak. A prominent member of the Vatican Academy for life, Elio Sgreccia, put forward the following ideas: assisted procreation breaches the natural right of the child to have only one father and one mother. Moreover, the natural right of the child to know who are their biological parents is violated.
These are not good arguments. First, marriage is presented as a natural institution. In support of this point, it is said that most of the constitutions consider marriage as an important institution within our societies. Needless to say, the fact that an institution is important within our societies does not make it 'natural.' To the contrary, the argument from constitution merely acknowledges that marriage is not a natural institution, but an artificial one, created by our laws.
Second, the right to know one's biological parents is parasitic to other more important interests. As such, we may agree that it is important to protect it. But we may just as well balance it with the interest of the donor in not disclosing his identity (His reasons for donating should not concern us). Moreover, the fact that the right to know one's biological parents is limited can hardly justify the limitation of the interest of the parents to conceive a child. Of course, the parents can still adopt a child instead of asking for assistance, as the Vatican commentator argues. I am theoretically in favour of that too, though I think that an individual should be free to decide whether to opt for adoption or assisted procreation. More interestingly, the very point of adoption undermines the strength of the argument from the natural right to know one's biological parents. For adoption procedures limit the disclosure of the identity of biological parents.