I would argue that Paine was not correct when he said that Europe, rather than England, was the parent of the American colonies. However, it makes sense that he would claim this as the claim would support the idea that the colonies should break away from England.
Paine based his claim on the idea that there were a very large number of people in the American colonies who were not from England. He said that America had become the “asylum” for people from all over Europe who wanted freedom. Therefore, America was not part of England as much as it was a part of Europe populated by those people who wanted to be free. You can see how this would be considered a factor for separation. If the colonies were not really connected to England by ethnic ties, there would be less reason to say that they needed to remain a part of the British Empire.
I, however, do not think that Paine was really justified in making his claim. If he were justified, we should find that the colonies’ population was largely non-English by the time that Paine wrote. This is not the case. According to this link, roughly 10% of the people in 1790 America (not long after Paine wrote) were Europeans from outside of Great Britain. There were many Africans, most of whom were slaves, but they were clearly not Europeans who came for freedom. It seems hyperbolic to say that England was not America’s “parent” if more than 80% of America’s population was English. Paine might have said this, however, because he lived in the Middle Colonies which were much more diverse than other regions.
Overall, then, I would say that Paine was probably not justified in claiming that America was more European than English, but his saying so makes sense because it would help to justify the colonists’ desire to break away from England.