It’s going to depend on when they’re going to hop parks. If it’s in the middle of the day they can keep their car where they are and just go to the Disney buses when the other parks’ buses start running.
You won't pass through the parking kiosk on the way out of your current park so your best bet is to ask when you get to your new park. But without proof that you've already paid for parking for the day, I don't know for sure what they can do.
Maybe you could stop by guest relations or a blue umbrella in your current park and ask for advice.
Go to Guest Services and explian your situation. I got to believe they will give you a pass to park your car at the next park you go to, as long as your explanation is credible.
We were not allowed on the boat from EP to DHS after Illuminations. The boats were still bringing people FROM DHS but, even though they were going there, they would not let us on. Had to walk to pick up our car.