The House of Commons disallows its MPs to also be members of foreign legislatures except for Commonwealth countries and, since 2000, Ireland. Why is Ireland a special case here?
The House of Commons Disqualification Act 1957 had disqualified from membership of the House of Commons those who were Members of non-Commonwealth legislatures. This clause was repeated in the House of Commons Disqualification Act 1975. This in turn was amended in 2000 to permit members of the Houses of the Oireachtas in the Republic of Ireland to sit in the House of Commons.
A footnote points out that dual membership was also permissible before 1957.
The same document has another instance where Ireland is a special case: a criminal conviction with a prison sentence in Ireland also disqualifies members.