A British subject born to British subjects on British soil moves to another patch of British soil and he is an "immigrant"?
You could not be more right. Ignoring that Hamilton came to "America" before it was "America", Parker is apparently also ignorant the fact that the first immigration restriction law wasn't passed until 1875. The Page Act of 1875 was the first law to create what could be described as an "illegal alien", and ended what could be described as America's open door immigration policy. Until then, anyone who landed on the shore, was welcomed - perhaps not eligible to be a citizen, but welcomed nonetheless.