The only way to do this (assuming you are not staying at the BC) is to valet park your car at the BC. Of course, you'll have to pay for that and if you add in a tip, you end up paying more for this "convenience" than if you drove to Epcot and parked there ($7).
If you are not a guest at the BC, you'll be given a 3 hour pass to put on your dashboard. This is designed to ensure there is enough parking available for guests of the BC -- otherwise, the lot would fill up with folks doing what you've asked about. (There are stories about cars being towed ... I've never seen that happen at BC, but then I've never spent my vacation scoping out the parking lot!)
My recommendation would be drive to BC, park, enjoy breakfast, and then drive over to Epcot. The exercise will do everyone good.