You might compare this with the concept of Track 1, Track 1.5, and Track 2 negotiations. When governments talk to each other, it is Track 1 and highly official. Academics and civil society are Track 2 and not official. Parties with government ties could be seen as something like Track 1.5.
In the european-union, there are different political parties in each country which organize themselves into groupings in the European Parliament, e.g. the EPP on the center-right, the PES on the center-left, ALDE for european-style liberalism, or the EGP for the Greens. The individual members of these groupings are quite diverse; just consider how long it took the EPP to throw out Fidesz.
Similar groupings and associations exist beyond the borders of the EU. PES membership overlaps with the Socialist International, and EPP membership overlaps with the Centrist Democrat International.
In many countries there is the political tradition to keep domestic politics inside and to stand together on the international stage. This is crumbling in the EU, where, say, German conservatives might make common cause with Austrian conservatives against the German socialists, and not see that as unpatriotic or illegitimate.
Another example from German history has a more antagonistic situation. After WWII, Germany was split into four occupation zones for the Soviets, US, UK, and France, and eastern areas seized by the Soviets and Poland. The three occupation zones in the West became the Federal German Republic (FRG) and the occupation zone in the East became the German Democratic Republic (GDR). The FRG always insisted that this division was just temporary, and in the meantime it pressured third parties to accept only the FRG as representative of Germany (the Hallstein doctrine). The two Germanies wanted to talk on practical and humanitarian issues, yet the FRG never admitted that their diplomats in East Berlin could be an embassy with an ambassador -- it had to be a mere permanent representation.
In the years before the permanent representation, when the FRG government tiptoed around anything resembling a recognition of the GDR, it was possible to have talks and letters between the Western SPD party and the Eastern SED party. These were understood to be somewhat substantial -- both parties played key roles in the respective states -- yet formally not state-to-state talks.
There are other countries, often authoritarian ones, where international cooperation gets labelled as foreign agents and suppressed.
And other cases where foreign parties are supported not because of a common ideology, but to destabilize another country.