For a current excitation, with a parallel tuned LC circuit in resonance the current flowing in the capacitor is theoretically infinite for perfect components. The same is true of the inductor theoretically.
Reality isn't perfect and usually the inevitable dc resistance of the inductor ensures the current might be a few mA to several tens of amps.
What is true is this; for decent high Q components the current flowing in both L and C can be many, many times higher than the current supplied by the power source.
For a voltage source the current supplied to the LC circuit may be theoretically zero at resonance whilst the current in C or L is determined by the voltage supplied and the component reactance