Parliament had no authority in the question of declaring of war, would never have refused to fund a war just declared, and would not have held a non-confidence vote in the first year of a war. Parliament had only the foggiest of notions as to what was going on and would have backed its PM and king in virtually any conceivable scenario.
If that were true, why bother with Grey's speech to the Commons on August 3? Why bother trying to persuade anyone if they'll just go meekly along in appropriating for any war you feel like declaring?
We all know why: No Prime Minister of that period is going to go to war, even with royal backing, unless he feels reasonably confident that he has a reasonable base of support in parliament (and hopefully his party).
If France is in no danger of being crushed - if Germany is taking a more docile public profile - you simply cannot say that Asquith's job doesn't get more difficult. Perhaps he *could* just form a coalition with Tories (a prospect he dreaded, by the way) and ram it through anyway. But public enthusiasm to beat the dreaded Hun in such a circumstance is going to be harder to fire up.