There's two parts to this - and I think the question concerned the first consideration, not the second, but... here's the two parts:
1) LEGAL rights? none. Not unless all custodial parents (for a minimum) have agreed to give you legal rights, via POA or whatever. If bio-parents have shared custody, or if the other parent has the custody rights... then unless other-bio-parent wants you to have legal rights, you don't get them. This really affects the ability to speak on the childs behalf at formal meetings (IEPs... ), make critical medical decisions (start/stop care, etc.), signing paper work... those kinds of things
2) PRACTICAL rights? some. For example - signing a kid out of school? As a parent, I can authorize all sorts of individuals to do this... grandparents, an uncle, the babysitter... so, yes, if your S/O can do it and gives the school (or whoever) for you to do it, then you're fine with these kinds of things. You can support your S/O at meetings, even be the "front-man" in the discussions... There's lots you CAN do.
But it gets really murky. Not everything is clear-cut.