NZ/Aotearoa now has an equal number of women in Parliament as men, for the first time. I don't know if any other country has achieved this. To make it more historical this is from the stuff.co.nz website:
"New Zealand women led the world in 1893, when women won the right to vote in the hard-fought suffrage battle.
It took another 26 years before they could stand for Parliament and a further 14 years before Canterbury woman Elizabeth McCombs was sworn in as the county's first female MP.
Former National Party MP Marilyn Waring is also among Parliament's firsts; the 15th female MP and youngest person to enter politics at 23 years old in 1975."

Is there a quota system or other mechanism that acheived this? I think that one or more of the Nordic countries does that.

No, we don't have any quota system. There is a Maori Party, but Maori can join any party they like, and our previous leader of one the parties, New Zealand First (known as the 'king-maker, since he often was part of a coalition) was Maori (though in name and appearance only really). 
Oh and I got that first post wrong - there are now more women (by one) than men.

Caro wrote:

NZ/Aotearoa now has an equal number of women in Parliament as men, for the first time. I don't know if any other country has achieved this.

At present there are four countries with more than 50% representation of women in the national parliament:
Rwanda         61.25% (and 38.46% in the upper house)
Cuba             53.22% (there is no upper house)
Bolivia           53.08% (and 47.22% in the upper house)
New Zealand   50.42% (again there is no upper house)

If considering representation in government as a whole, ie both upper and lower houses where appropriate to the political system used, Mexico does nearly as well as Aotearoa/New Zealand with 48.2% women in the Chamber of Deputies and 49.2% women in the Senate.

Interestingly Switzerland, mentioned above for only giving women the vote and allowing women into parliament in 1971 (and it was not until a 1990 decision by the Federal Supreme Court of Switzerland that women gained full voting rights in the final Swiss canton of Appenzell Innerrhoden) now has better representation by women than many other countries, including the UK and France, with 41.5% women in the Federal Assembly.

A belated thanks to you, MM, for those figures. The three countries above Aotearoa/New Zealand are very interesting, not being countries that would automatically come to mind when you think of women's rights. Rwanda is a country I think of fondly mostly due to Alexander McCall Smith's novels set there. 
And Cuba I just think of as Fidel Castro and his power. 
And I don't really think of Bolivia at all, though checking wikipedia it seems to have had a very interesting history.

Bahrain, the island state in the Persian Gulf, is an interesting case. On paper it is a constitutional monarchy with 50% representation by women in the upper house (which is where all the real political power resides as it exercises a de facto veto over the lower house, the Council of Representatives, who can only draft legislation if so directed by the upper house). However these 20 women councillors are not elected but are simply nominated by the King and indeed most of them are actually members of his own immediate family. Women only got the vote and the right to stand for election to the lower house in 2002 and there are no women representatives for any of the Islamicist political parties; only for the minority jewish and christian groups. There have been some notable nominations of Bahraini women into high profile government positions, for example when in 2006 Bahrain was elected head of the United Nations General Assembly, it appointed a woman, Haya Rashid Al Khalifa as the Assembly's President, but I suspect this was mostly for show (she was after all a close cousin of the king) and such high-profile cases are not really representatitve of the status of Bahraini women as a whole.

Alexander McCall Smith's 'The No. 1 Ladies' Detective Agency' series of novels is set in Botswana, a country which although ranking highly on all democratic indices - having a consistent record of uninterrupted democratic elections and the lowest perceived corruption ranking in Africa - does nevertheless currently have only three women in its 57-seat National Assembly (it has been higher in the past although not by much). But at least this situation is widely recognised as a problem with steps being taken to address it.

It seems that Rwanda's position of having a majority of women in government came about because of the national conflict and genocide that killed 800,000 Tutsis in 1994: the percentage of women in the legislature went from 18% before the conflict to 56% directly after. Two pieces of legislature enabled and supported women into leadership positions: a Security Council Resolution urged women to take part in the post-conflict reconstruction and the 2003 Rwandan Constitution included a mandated quota of 30% reserved seats for all women in the legislature. Of the 24 women who gained seats directly after the quota implementation in 2003 many joined political parties and chose to run again, although it took almost 10 years after implementing the gender quotas for Rwanda to reach a level of female representation which is amongst the highest in the world.

Oh dear, once again I have made a basic mistake, putting AMSmith's Botswana series into Rwanda, even though I have read most of them, though not very recently. He came to Dunedin once and I have signed copy of one of his books. 

It's interesting that Rwanda has mandated quota: do other countries with high representation of women have that too? I know NZ doesn't. I can't see the latest figures for NZ mayors (we just had local body elections last month) but the time before there was 27%, though it was higher in the cities than the provinces or districts.

Although I was aware that women getting the right to vote in New Zealand hadn’t equated to having the right to stand for election, I hadn't appreciated that it had taken 40 years before the first woman was elected to parliament. 1933 does seem incredibly late for the country’s first female MP. Even then it seems that Elizabeth McCombs was the widow of the sitting MP who had just died when she was elected to replace him in a by-election. She then served less than 2 years herself before dying and being replaced in another by-election by her son.   

It's also surprising to learn that New Zealand still had had only 15 female MPs in total by 1975. This compares, for example, with Finland where 19 women MPs were elected at the first time of asking in 1907. Worth noting is that Finland was still a grand duchy within the Russian empire at the time. The empire was going thru a period of liberalisation following the 1905 revolution which had been triggered in large part by Russia’s humiliating defeats during the Russo-Japanese War. The electoral reform introduced by the Finnish assembly had received imperial assent from the tsar (or rather grand ducal assent from the grand duke, with the tsar and the grand duke being the same person). 

With regard to the use of gender quotas in elections, then the Soviet Union is said to have employed these to some extent. The Ukraine today certainly has an electoral code which mandates that any given political party must include at least 30% of women on its list of candidates.