I don't really see it either. At best one can say that there may be some internal political forces in Turkey that can benefit from destabilization. In particular the 'old' secular parties and movement that want to undermine the popularity of Erdogan and the AKP. Much of its popularity is of course also due to the good economic times in Turkey lately. But of course, this strategy would only work if people would not blame the protesters for any economic damage.
In reality foreign parties care mostly about a stable and economically vibrant Turkey with which it is easy to trade, and that is exactly what the AKP has achieved the last 10 years or so. So I don't see why anyone foreign power would want to undermine that.
No, this is clearly an internal Turkish conflict. It is based on the exact same division that is so prevalent throughout the Muslim world: the Islamic vs. secular ideological spit. This together with the Sunni <-> Sh'ia split is clearly one of the biggest destabilizing factors in the region.