If you don't mind riding the buses:
- Park at the MK parking lot in the a.m.. (which is actually at the TTC)
- Bus direct to DAK for the day
- Bus from DAK direct to Poly
- Monorail or boat to MK
- Monorail or ferryboat to TTC & parking
You will pay once to enter the MK parking lot. All WDW buses, boats, and monorails are free to all.
Plan on about 45 minutes from the time you park your car to get to DAK. That includes walking from the car to the bus stop at TTC, waiting up to 15 minutes for the bus, a 15 minute bus ride, and a short walk from the bus stop to the DAK gate.
Allow an hour and a half to get from DAK to Poly by bus. The bus stops at Contemporary before Poly, so the ride is about 10 minutes longer. You'll be leaving DAK right after the parade, so you might have to wait an extra 15 minutes for a 2nd bus. Plus you want to get to the luau a little bit before the start time.
If you decide to park at DAK and then drive to TTC, the drive time is about 15-20 minutes. But -- you need to allow time to ride the tram from the DAK gate out to your car, ride another tram from the MK parking lot into the TTC, and then walk to Poly (or ride the Resort Monorail 1 stop). I'm not sure how often the trams run in from the parking lot to TTC at that time of day; I know I have waited as long as 15-20 minutes for a tram out to the lot when I've left MK in the afternoon.
If you decide to drive right to Poly after DAK, you will either have to move your car when you go to MK, or park valet and pay a separate parking fee. That will cut the time for the tram and walk on the Poly end, but you still need to allow time to get to your car in the DAK parking lot.