The Social Democratic and Labour Party is a Northern Irish party, which is moderately Left wing and supports union with the Irish Republic only by peaceful means and via a referendum, with the North retaining some links to the UK. It’s supporters are drawn from both Catholic and protestant communities, but both Christian denominations in Northern Ireland are highly traditional and tend to oppose homosexuality as a sin. The SDLP thus has to tread very carefully over the issue and does not make it’s position clear, in order to avoid an open confrontation with religious leaders.
The Liberal Democrats in England USED to be known as the Social and Democratic Labour Party in the late 1980s, although they changed their name in 1992. This party DOES support same-sex marriage.