I plan to take a day trip from Split or Dubrovnik to Korcula. How long does it take to go on the ferry or Catamaran? Also, is there safe parking to leave my car at either ferry dock?
My response to your other query gives details of the catamaran/ferry options. There aren't car parks at the docks - you will need to find a car park if you are going to be away for the day. Korcula has more options from Dubrovnik for just a day trip - it's a bit far from Split.
You might be able to pick up a commercial tour from Dubrovnik that includes Korcula plus Ston and other sights on the peninsula, but to do it by Public Transport may not work. There just is not ferry options that allow getting there, time to see things, then return the same day.
Since you have a car, I suggest driving to Korcula. The drive up the peninsula is pleasant, Ston a nice stop midway, nice beach at Orebic, then for Korcula either take a pedestrian ferry across the strait or you can take the frequent car ferry.
The Kapetan Luka catamaran goes south in the mornings and north in the evenings. From Split, you could do a day trip where you leave at 7:40AM, arrive at Korcula at 10:00AM, and then leave at 6:20PM, arriving back in Split at 8:50PM. That would give you around 8 hours or so in Korcula. Obviously, you can't use KL from Dubrovnik for a day trip as the catamaran is going the wrong direction.