Perhaps a more intuitive way to look at it is to just consider the reactances in parallel.
For the inductor, the reactance is Xl = wL where w is omega - 2 pi f. We can see that the reactance increases with increasing frequency. Conversely, the reactance decreases with decreasing frequency. For the parallel circuit operating below resonance, the inductive reactance dominates the circuit.
For the capacitor, the reactance Xc = 1/wC so the reactance decreases with increasing frequency and increases with decreasing frequency. For the parallel circuit operating above resonance, the capacitive reactance dominates the circuit.
In either case, the farther the circuit is from resonance, the lower the overall impedance and the higher the current flow.
Pick a really big number for w: Xl will also be a big number and Xc will be a small number. For a big number, the capacitor dominates the parallel combination.
Pick a really small number for w: Xl will also be a small number and Xc will be a large number. For a small number, the inductor dominates the parallel combination.