Women in same-sex relationships
Women in same-sex relationships have the right to receive artificial insemination on the same level as different-sex couples. Marriage is not a requirement, but the relationship must have shown some stability. The co-mother's consent is required.
If a woman in a same-sex relationship receives artificial insemination and the child can learn the biological father's identity when coming of age (18), her partner can become a co-mother from birth. She will then have the same rights to parental leave as a father does. If the requirements for co-mothership are not met, the couple has to be married and go through a step-child adoption procedure. There are some guidelines from the Ministyr of Children, Equality and Inclusion saying that this procedure should be gone through as quickly as possible, at the earliest it can be done 8 weeks after birth since the law says that the mother can give her consent to adoption no earlier than two months after birth. (the Child Act (Barneloven) chapter 2)
Men in same-sex relationships
As for men in same-sex relationships having children is a lot more complicated. Formally they have the right to adopt, and then everybody has the same rights to parental leave no matter their gender. In practice, it is impossible to adopt from abroad, and there are very few Norwegian children for adoption, so the right remains almost illusory. Some go abroad and have children by a surrogate. The non-biological father then has to go through a step-child adoption procedure.
The general rule, providing the mother has worked six out of 9 months before birth, is as follows: The parents have 46 weeks
with full salary up to approximately 56.000 Euros, or 56 weeks with 80% salary, including 3 weeks before birth, which they
can share as they wish with the following restrictions: The first six weeks after birth can only be taken out by the mother,
and 10 weeks of the remainder period is exclusively the father's. (The Social Security Act (Folketrygdloven) chapter 14)
When the couple has to go through at step-child adoption procedure, the rules concering parental leave only enter into force when the co-parentage is formally recognized. Since the mothers consent is required, but can only be given two months after birth at the earliest, this can happen no earlier than two months after birth. This rule applies also with surrogates.
Once the adoption is in order, the co-parent has the same rights to parental leave as a father, but only for the remainder period of leave. Most often the step-child adoption procedure appears to be done as requested by the ministry.
For an overview over the situation in various countries, you can vistit ILGA Europe's website.