There are two ways to lower the deficit: increase revenue or lower spending.
As you say, neither party does particularly well at lowering spending.
The moderate Republican approach to raising revenue is typically but not always through regressive taxes. The general theory since Reagan (and before) has been taxing the upper brackets hurts economic growth. The ideal of this is a flat tax or “fair tax.”
The centrist Democrat approach to raising revenue is typically but not always through progressive taxes. The general theory is that wealth inequality is overall bad, and since purchasing power is more or less logarithmic in dollars, the upper brackets suffer relatively less from higher tax rates. The ideal of this is the post-war 90% top bracket.
Of the two, I think only the second offers any real hope of deficit reduction. The middle and lower classes are already tapped out, metaphorically speaking. Household debt is already frighteningly high and the savings rate very low despite relatively high interest rates. If the goal really is to reduce the deficit through increasing revenue, I think it’s likely that revenue must come from the upper classes in one way or another.
Of course this is all written in very broad strokes and there are a hundred nitpicks one could make on this general point of view, but this is a forum comment and not a dissertation on political economics.