I've always parked at Porta Catania parking structure. Save the address below in your GoogleMaps.
https://maps.app.goo.gl/7iAm479sUS89PsEz7
The image below shows what it looks like. It's huge and I've never had problems finding parking even in June:
https://maps.app.goo.gl/w51YGmDG14FJkvdU6
Once you park, you go to the proper floor with the elevator and you are at the town center level, just a short walk away.
There is another parking structure at Porta Pasquale, address below:
https://maps.app.goo.gl/tRe2acGg1uGnqYP77
There is no reason to pay 500 euro to a taxi driver when you can drive on your own for 40 euro of gasoline round trip (plus parking fees of maybe a couple of euro per hour).