This really ought to be phrased something more like "According to the best current model of the universe, there was no time until the big bang occurred. However, this model has known flaws; it is the best because we haven't got anything better, not because it is actually correct. Many competing models have been proposed, many of which include some form of time before the big bang." It really should not be stated as unassailable fact that time started with the big bang. It isn't even unassailable fact that the big bang must have erased all information about the previous universe.
In the early universe, before the plank epoch when we had particles of plank length or less, you know, things with a schwarzschild radius? The entire universe made of things that are black holes, nothing but black holes? Those time-ending particles that disappeared in their own hawking radiation? The first early 10^−43 seconds, where time didn't actually exist? Those were the good old days.
The first early 10^−43 seconds, where time didn't actually exist?
If time didn't exist, then how did the universe know when those 10^-43 seconds were up?
Seriously, I think this discussion is proceeding at a level which probably shouldn't be attempted by non-cosmologists. We all probably have a bunch of misconceptions we've picked up from popularizations about the Big Bang, and would only embarrass ourselves if any cosmologists happen to show up.
You didn't. Sorry, my phrasing was poor. You said that time didn't exist at the earliest part of the Big Bang, and I was trying to clarify that time didn't exist "before" the Big Bang but did immediately after.
In the earliest part of the Big Bang time did exist. The Big Bang created time.