Lamia, I generally concur with both KellyM and Lilairen. If you do review this thread, you will note my early attempts to avoid advocating for polygamy, but instead to refute factual errors made by other posters. In each case, I took great effort to provide citations to support my statements. Despite my intent, I did end up making some advocacy statements, and I’m now prepared to debate them.
Let me state unequivicably that I would never advocate any marital system that failed to provide for equal rights among all participants. Consent is crucial in bringing in additional spouses (but no veto power, without consent, termination of the original marriage contract would have to occur before another could be entered).
I’ve already described some of the details, as have others. Now I’ll address your specific query related to parental rights. When a woman has a child, she has the option of specifying the father for the birth certificate. Only the biological mother and the named father (if named) have presumed parental rights (just like m-f marriage or a birth out of wedlock, for that matter). The named father could contest the obligations through a paternity test (science has advanced to make this much more practical than it would have been only a few decades ago). If a paternity test proves another father, that father would then be obligated with the parental rights.
For the most part, nothing very different from the situation today (remember when I said “it need not be any different”?), except that in marriage today, the husband is automatically presumed to be the parent, regardless of actual paternity (an inequity, IMHO).
Under polygamy, I would expect to see the government allow other spouses (with the unanimous consent of those with parental rights) to adopt children in the household and accept parental rights. Likewise, with the consent of those involved, I would like to see spouses divorcing (terminating their marriage contract) claim or disclaim parental rights. As with any divorce, the courts should be charged with looking out for the best interests of the children, and have the ability to deny any disclaimer (you can only give up your parental obligations if the courts agree it to be in the best interest of the child).
Now, unlike many of the gay marriage advocates, I’m not hung up on terminology. If enough people are hung up on “marriage being sacred”, and worried about expanding the definition of marriage, then let’s use different terminology. But whatever is in place, let it apply to all equally. I’ve already stated in this thread that I think government should get out of the marriage business, and leave marriage to the churches. The equivalent legal protection I would advocate can be achieved through domestic partnership law (similar to what you describe as a civil union), much like what is being discussed in the American Law Institute’s review of these types of issues. For an interesting technical read regarding the legal issues involved generally, see this link (the discussion on polyamory issues starts in section IV, but the first sections provide useful context). If you seriously want to consider specifics of law, reading this would be a pretty good start. It also provides a pretty good explanation why simply removing archaic cohabitation laws might be less than optimal.
I can’t find a source online, but an even better description by the same author can be found in the Harvard Civil Rights-Civil Liberties Law Review, Winter 2001, in an article by Martha M. Ertman titled, Marriage as a Trade: Bridging the Private/Private Distinction. When you ask questions about if A is married to B and C, but B and C have no relationship, you may find it useful to consider existing business law to think about the implications. IOW, yes, I believe that situation could exist without any serious problems, provided B gave consent for A to marry C. In a case like this, if A died, his/her assets are split between B and C, and no further legal relationship between B and C would exist.
Stonebow,if you are particularly interested in the Rubin and Adams study, it was published in the Journal of Sex Research titled, Outcomes of Sexually Open Marriages. I also won’t advocate that polygamy/polyamory is “better” or “more advanced” than monogamy, but I do advocate that social acceptance of poly is better than shunning it. I fully believe that most people are better off in monogamous relationships, just not everybody. I also believe that many of the problems associated with the Mormon Fundamentalists are exasperated by the illegality of polygamy (much like how the War on Drugs has created a huge black market that causes many of the problems now blamed on drugs).