[adhesive_wombat]

No. Getting to the Sun requires scrubbing off nearly all the orbital speed of the Earth (30km/s), or the payload just orbits at a smaller radius.

The delta v to the Moon is something like 7 times less than to the Sun (3 vs 20ish).

The delta v need of the Parker Space Probe, which still "only" gets to 8.5 solar radii is so high it will use 7 Venus assists to get it close enough (it can't do big Jupiter assists because the solar panels it would need at Jupiter wouldn't be able to fit behind the sun shield at perihelion, and they didn't want to give it an RTG because they're saving plutonium for future missions).

You can eventually hit the Sun with enough Venus assists and Earth assists, for a total delta v of under 4 km/s, but it'll take a very, very long time, and your nuclear waste will be doing Earth flybys until finally it hits the Sun. Also it might be tricky to get the ball of sun-melted radioactive slag to do accurate assists after the sun melts it on the last few close encounters.