parallel rlc voltage source
Use the Laplace transform. Several forms of the voltage across the parallel combination are possible, depending on component values. But this voltage is always zero at t=0

If the system is underdamped, this voltage is an exponentially decaying sine term, hence equal to zero at t=0

If the system is overdamped, the voltage is the difference of two exponentials and, again, zero at t=0

For equal roots, the voltage is of the form \$te^{-at}\$, so equal to zero at t=0

The above assumes that all component values are non-zero, and zero initial conditions. You're question is contradictory on initial conditions - 'no energy storage' and 'there is current in L1 at t=0- '