Registered domestic partners can claim a qualifying child(ren) as a dependent on their taxes but the IRS has rules for how to go about doing this. 

If a child is a qualifying child of both parents who are registered domestic partners, either parent, but not both, may claim a dependency deduction for the qualifying child. If both parents claim a dependency deduction for the child on their income tax returns, the IRS will treat the child as the qualifying child of the parent with whom the child resides for the longer period of time. If the child resides with each parent for the same amount of time during the taxable year, the IRS will treat the child as the qualifying child of the parent with the higher adjusted gross income. (IRS Website - Q&A 3)