In other words, unless it can be shown that a phenomenon cannot be explained unless there is a God, then we have no reason to think that there is a God. God would complicate the issue rather than work towards resolution.
First of all, let me say that there is a difference between parsimony and simplicity. There are many simple theories which are not parsimonious. For example:
|"The planets and stars are stuck to orbs around the Earth."|
But even though the idea of a heliocentric solar system is vastly more complex, it is still more parsimonious because it leaves fewer questions unanswered.
This principle is referred to as Occam's Razor. The idea that the hypothesis that requires fewer assumptions is preferable.
That's actually one of the first weaknesses with this argument. It's presents Occam's Razor as an absolute law, when it is actually just a "rule of thumb".
This is important because there are many, many examples of Occam's Razor stifling or delaying scientific advancement. For example, appeals to simplicity were used to deny meteorites, ball lightning, continental drift, DNA, atomic theory, and reverse transcriptase.
In addition to denying ideas that turned out to be true, Occam's Razor is also used to assert ideas which are false. For example, scientists thought that there must be some medium, Aether, out in space which allowed light waves to propagate, because it was less parsimonious to postulate wave propagation in a vacuum. At the time, all known waves propagated through a physical medium. We now know this to be false.
So the Argument from Parsimony fails immediately just because explanations sometimes reach beyond available data.
Second, even if the argument was successful, denial of God would bring us no closer to understanding what caused the Big Bang or life on Earth. Even Richard Dawkins admits the possibility of an origin involving intelligent beings (3:10).
For example, imagine yourself holding the end of a long chain. The other end of this chain is not in view, but when you pull on the chain, it it secured. If we were to say that there is nobody holding the other end of the chain, would we be any closer to finding out what is holding the other end? For that matter, do we have any good reason to think there is no one holding the other end? Sure, there may be no evidence that there is, but is there any evidence that there isn't? Like Dawkins, should we at least be open to the possibility of some intelligence on the other side?
Would we be making any fewer assumptions if, instead of a person on the other end of the chain, we instead said that it was fastened to some post or bolt in a wall? Or would we then need to explain how that post got there, ad infinitum?
So even if the argument is successful, which it isn't, then it still just doesn't do us any good. It wouldn't even necessary show that God is the less parsimonious hypothesis.