There are currently Pirate parties in 40 countries and Green parties in 52 countries. But is there an organization which owns the rights to the Pirate/Green party brand? Or can anyone go ahead and start his own Pirate party without any approval from the original founders?
International political movements
The Global Greens is an international organisation of Green parties; it has sub-groups such as the European Greens.
The Pirate Parties International plays a similar role for national Pirate Parties.
Both these movements share some similarities with the long history of international socialist cooperation, dating back to at least the late nineteenth century.
International political movements could refuse membership to a particular party; but whether a party can use the Green (or Pirate) name is determined by national election laws.
Example of national law on party names
The UK has had a few mischievous attempts to create confusion by forming parties with very similar names. In 1994 a candidate created the Literal Democrat Party, easily mistaken for the established Liberal Democrat Party. This practice was banned by legislation in 1998. If I want to create the "Anglo-Welsh Green Party" as a separate organisation from the (already existing) Green Party of England and Wales, the law is unlikely to allow it; but before 1998 this might well have been possible.
International (lack of) enforcement
Internationally, there is no real enforcement of consistency on party names. It is common for different countries to have parties with the same or similar name, but quite different views.
For example, each of Japan, the UK, and Russia has a party named the Liberal Democrats, but their history and policies are wildly divergent. The Japanese party has existed since the 1950s; the UK party since the 1980s; the Russian since the 1990s.
There has been no attempt by the older Japanese party to force the UK party to change its name; nor is there any indication that the UK authorities would take it very seriously if it tried. The (centrist, human-rights-oriented) UK Liberal Democrats may be unhappy about sharing a name with the (right-wing nationalist) Russian Liberal Democrats, but there is nothing much they can do about it.
There's no common ownership of the brand. Some Pirate Parties have sought national ownerships over their brands, not least in light of the fact that the brands have been used maliciously.
Here in Iceland the idea came up when a different party formed a regional coalition with a few pirates, and they attempted to run under the Icelandic term for pirates.
Regarding whether anyone can run using the brand. The short answer is yes. And the fact is, the parties differ quite a bit from one another between countries according to regional needs.