co-parent
Of a set of parents who were previously married or in a relationship with each other, to jointly raise their child(ren). My ex-boyfriend and I co-parent, so the kids will be with him this weekend. John co-parents with his ex-wife.
co-parenting
Of a set of parents who were previously married or in a relationship with each other, the act of jointly raising their child(ren). I know co-parenting isn't always easy, but at least the kids get to spend time with both of you.