If you have children who live with you and your bidie-in, and your bidie-in has accepted the children as part of the family, then your bidie-in is the children's stepparent.
Co-parent is a less widely recognised term than stepparent. As the term is used within the LGBT community, when a couple decide together to have a child who would be biologically related to only one of them, the non-biological parent is the child's co-parent. (Outside the LGBT community, co-parent is often used as a verb to mean two parents who are separatiing but who intend to take joint responsibility for their children.)
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