It's historical. In consumer theory you derive the individual demand curve by maximizinging your utility function given prices and income. So price is the independent variable, quantity the dependent. In 'the theory of firm' you get the individual supply curves from maximizing your revenue function where you adjust quantity by given factor prices and given price of the good you produce.
So the agents in crazy neoclassical complete market micro are always price takers. Given some prices, income, costs- what quantity would you choose? That gives the curves. Add them all together, you get the aggregate supply and demand curves.
On a supply-demand curve it's supply and demand that are the determining factors, both quantity and price are results.
Well, there is shifting the curves and moving along the cuves. Or in eco lingo, exogeneous and endogeneous variables.
You first need demand and supply curves to begin with. They are formed in micro theory as described above.
Actually price and quantity are mutual variables; each can be determined by the other.
Only in the real world, not in your average microeconomics textbook.
Allright, in oligopoly theory you have this obvious interdependence, but the nice demand and supply curves are gone.