Democratic Unionist Party = an Páirtí Aontachtach Daonlathach. Treated as a definite noun construction, with a definite article. Genitive: an Pháirtí Aontachtaigh Dhaonlathaigh.
Sinn Féin. The same in Irish. Treated as a proper name, with the genitive form Shinn Féin.
Social Democratic and Labour Party. = Páirtí Sóisialta Daonlathach an Lucht Oibre. Treated as a noun followed by a definite genitive, which means that the genitive form is Pháirtí Sóisialta Daonlathach an Lucht Oibre.
Ulster Unionist Party = Páirtí Aontachtach Uladh. “Uladh” is the genitive form of “Ulaidh” = Ulster. Thus, the translation suggests “the Unionist party of Ulster”. Being a proper name, Ulaidh is inherently definite. Thus, it is illegal to add a definite article before Páirtí. The genitive of the party name is thus Pháirtí Aontachtach Uladh. (A thinkable alternative translation would be “an Páirtí Aontachtach Ultach“, where “Ulster” would be translated with an adjective. In this version, an article would be necessary, and the genitive would be “an Pháirtí Aontachtaigh Ultaigh“.)
Alliance Party of Northern Ireland = Páirtí Comhghuaillíochta Thuaisceart Éireann. Éire = Ireland is a proper noun, thus inherently definite. Its genitive form can take an article or omit it, but both ways it is inherently definite. Thus, the construction Tuaisceart Éireann has the genitive Thuaisceart Éireann. In the construction Páirtí Comhghuaillíochta, Comhghuaillíocht is not a proper noun, but a noun in genitive (-a) modifying the word “páirtí“, and “páirtí comhghuaillíochta” is an alliance party. However, when Northern Ireland is added, it becomes definite: Páirtí Comhghuaillíochta Thuaisceart Éireann. And its genitive form is Pháirtí Comhghuaillíochta Thuaisceart Éireann.
Traditional Unionist Voice: an Glór Traidisiúnta Aontachtach, genitive an Ghlóir Thraidisiúnta Aontachtaigh. I am not terribly happy with this version, because “Glór Traidisiúnta na nAontachtóirí” (“the Traditional Voice of the Unionists”) or “Glór na nAontachtóirí Traidisiúnta” (“the Voice of the Traditional Unionists”) would sound better in Irish.
Green Party in Northern Ireland = an Comhaontas Glas i dTuaisceart Éireann, genitive: an Chomhaontais Ghlais i dTuaisceart Éireann.
UK Independence Party: Wikipedia provides a translation that is definitely wrong. I suggest Páirtí Neamhspleáchais na Ríochta Aontaithe, genitive Pháirtí Neamhspleáchais na Ríochta Aontaithe.
Progressive Unionist Party: an Páirtí Aontachtach Forásach, genitive an Pháirtí Aontachtaigh Fhorásaigh.
People before Profit Alliance: The suggested translation is Comhghuaillíocht an Phobail roimh Bhrabús, but I am not entirely happy with this, as it means “The People’s Alliance before Profit”. Besides, pobal means rather “community” than “people”. However, the genitive form of the suggested translation is “Chomhghuaillíocht an Phobail roimh Bhrabús“.
Animal Welfare Party: I’d suggest Páirtí Cosanta na nAinmhithe, i.e. the party of defending the animals. “Animal welfare” would be “leas na n-ainmhithe“, I guess, so a word for word translation would be Páirtí Leas na nAinmhithe.
Note that theoretically speaking, “(a) Unionist” as a noun is the same as “Unionist(ic)” as an adjective, i.e. Aontachtach. However, the noun Aontachtóir is usually preferred. The noun for any kind of nationalist is basically Náisiúnaí, but in Northern Ireland contexts Náisiúntóir is commonly preferred.
The noun for Loyalist is Dílseoir. However, we also need the adjective for such expressions as “Loyalist terrorists”. My best bet would be “Dílseach“, i.e. Loyalist terrorists = sceimhlitheoirí Dílseacha. Loyalism is Dílseachas or Dílseoireacht.