I would probably park at the park you were going to end up the night in. Just be aware of closing times, obviously. It is easy to get around in the car if you prefer. Your parking pass is good at all parks so you can go to MK in the morning and EP in the evening on one parking pass.
We always stay off property and drive and we’ve never had an issue.
Be very aware of which roads have tolls. Florida, like other states, is moving away from tollbooths and toward electronic toll monitoring. If you go onto a toll road and don’t have a SunPass they will find you. We drove to Miami to get on a cruise ship this past summer and ended up on a toll road because of a traffic jam. A few months later we got a bill for the toll. If you plan to use toll road a lot you can buy a SunPass and then you won’t have to pay the “convenience fee” if they bill you later. https://www.sunpass.com/index