Here's the Op-Ed in question: http://www.nytimes.com/2016/07/04/opinion/did-a-fear-of-slave-revolts-drive-american-independence.html
/u/sowser can chime in here (it's their area of expertise, after all) but I find Parkinson's argument persuasive, and it's not really a new one. Parkinson, Gerald Horne, Gary Nash, the Blumrosens, Simon Schama and Cassandra Pybus (to name just a few) are part of a group of historians who have really been doing a good job of explaining "why tens of thousands of Americans ran away from the Declaration of Independence," as Parkinson explained in a 2007 series of book reviews in the Journal of the Early Republic.
The core idea is that Americans and Caribbean planters depended upon black, chattel slavery in a way that no other part of the British Empire did. As the heart of the Empire expressed antislavery sentiment, particularly in the courts, and seemed to be moving toward some kind of abolition. The steps made in the 1770s were small, but they were there, and because these planters depended on slavery -- indeed, most of their wealth consisted of the value of enslaved human beings -- they were extraordinarily concerned about any changes to the existing slaveholding system.