Yes National Parks are out of bounds except with permits
Yes, near impossible here for hobbyist pilots, even commercial require permits with fees.
Many councils have no drone rules so taking off and landing is technically out, fly over ok, but that kind of makes it hard and pointless anyway.
If you do it quietly and while not disturbing other park users (ie ensure no one else around) then you won't get disturbed in general, I am yet to see any 'no drone' signs in any parks in such council areas.
I like the coast down Hallett Cove way, the track along the cliff tops is awesome to fly from for sunset, and it winds in and out of the Hallett Cove Conservation Park, so as long as you fly from outside that you're good there.
There are also boardwalks down to the rocks to fly from low down at sea level.
Marino Rocks boat ramp (no longer used) is also a nice spot.
Want to go a bit further ?
The coast south along the Fleurieu is packed with beaches to fly, Rapid Bay / Second Valley are 2 great spots.
Big wind farm down closer to Cape Jervis, south coast towards Victor Harbour, some great spots, flew out to the West Is 600m one way a while ago, a marine reserve so just fly off the landfall a little.
Further around to coast to Goolwa, loads of great coast along the way.
Always hard to find quiet places at some of those weekends, weekdays or really early is easier.