In the earliest part of the Big Bang time did exist. The Big Bang created time.
Edit:
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.
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.