Like others have said, you can use Disney busses, monorails, and boats for transportation. However, during busy periods (park closing, other times during peak seasons), you may be denied access. That being said, though, I have never been asked to show resort id to use Disney transportation.
justmitype - If you want to start your day at MK, you have three options -
1) Park at Epcot, then monorail to the Ticket and Transportation Center and either boat or monorail to MK. Later in the day, simply monorail back to Epcot.
2) Park at MK, then monorail to Epcot, then back to the TTC at night.
3) Park at MK, then drive to Epcot later in the afternoon. Honestly, this option would probably be just as quick as the first two.
Also, once you have paid for parking, you should get a paper showing proof of that. That paper allows you to enter any theme park parking lot without paying again for that same day.