This is a fallacy, of course. And there are sound arguments against it here, here, and here. Ask yourself, would any laws about marriage have to change for same sex couples to marry? The answer is, "no". Marriage law is based on a couple. Then ask yourself, would any laws have to change for multiple partners to marry? The answer is, yes. For example, what happens if one person in a polygamous marriage leaves. Are the others still married? How much property does the leaver get to take?
The UN condemns polygamy, stating
Polygamous marriage contravenes a woman's right to equality with men, and can have such serious emotional and financial consequences for her and her dependants that such marriages ought to be discouraged and prohibited. The Committee notes with concern that some States parties, whose constitutions guarantee equal rights, permit polygamous marriage in accordance with personal or customary law. This violates the constitutional rights of women, and breaches the provisions of article 5 (a) of the Convention.
There's a new article by Elizabeth Abbot on polygamy that is wrestling with the issues raised by fundamantalist Mormons in Canada, and religious freedoms. It asks rhetorically whether one has "religious freedom" to practice abuse. Abbot concludes
[Le]galizing [polygamy] is not ultimately in the same category as granting a pastor the right to express his loathing of homosexuality, or as legalizing gay marriage. While much has been made, in particular, of the parallel between sanctioning same-sex unions and sanctioning polygamy....the outcomes couldn’t be more different. The former brought people into an existing system of rights; the latter poses a significant threat to that system. And that’s probably our cue, as a liberal society, to hold our noses and draw the line.