We also take our own car whenever we go. We usually stay off-site also. While park hopping, we just leave our car in whichever lot we park in in the morning.
We have gone to AK in the morning, parked our car, left in the late morning, drove our car to MK or MGM and parked there. But the Disney Transportation is Free and available.
The choice is yours.
Say we do MK during the morning. We park in MK parking lot. The park closes early so we head on over to EPCOT via monorail. When EPCOT closes, we hop on the monorail back over to MK, and then pick up our car.
If you do MGM, there is no monorail service to/from that park. You just take one of the specially marked buses that say Magic Kingdom (or whichever park you're parked at). It will bring you back over to that park and then you pick up your car.
Also, if you know which park you want to be in during the evening, you can park in That parking lot & take transportation over to the park you want to spend the morning in and vise-versa.
Of course, Disney transportation (buses) are not the quickest way. If you spend money for parking in one park for the day, and you choose to drive over to another park in the afternoon, you do not have to pay for additional parking. You pay for the day no matter where you want to park. You have your parking ticket from the last park and the next parking attendant will honor it, as long as it is the Same Day.