Yep, it doesn’t seem to be a common position, for sure.
I have heard of the American Solidarity Party and like some of their viewpoints, but I think that any consenting adults should be able to have a relationship (and the corresponding legal benefits). I also don’t like the religious motivations behind many of the arguments of the ASP; my views against abortion are secular in nature, and I think the religious arguments probably hurt the pro-life movement as a whole.
Political parties like to pretend they have monopolies on certain positions, but that's just wrong. You'll find lots of gay rights-supporting Republicans and a whole mess of pro-gun Democrats. Some voters are rational enough to realize that they should put more weight on the issues that directly effect them than the nice-to-have emotional stuff that doesn't.
Without this, it makes absolutely no difference what the platform is of the Greens, the Libertarians, the Solidarity Party, or any other 3rd party: they can only work as spoilers in an election.
Get us to an electoral system in which the winning D or R candidate is faced with knowing that 40% of their vote came from people whose first choice was the Communo-Anarchist Party, and another 10% from people who preferred the "White Power Now" party, and you're going to see a totally different sort of politics.
There is a minor party called the American Solidarity Party that is a little more of a mix, similar to Christian democratic parties in Europe. They oppose abortion and the death penalty but support universal health care, environmental protections, stricter regulation of financial markets, etc.