as the PP said the only parking is at the resorts and some times they limit you to only 2 hours
I would just plan on parking at the main entrance or if you do not want to do this park at HS and take the boat over it will take you longer but nice and relaxing
You can valet park at the Epcot resorts but that will cost you just as much (if not more with tipping) as parking at the front entrance. Your walk time to get to the Gateway will be about the same as the front entrance and you won't have the option of the tram.
Or you can park at Swan/Dolphin. They are all pay lots even if you are staying there. You just park and when you leave you get a ticket and pay. I think it's around $23 for the day so it would be less to either pay at HS and boat it or pay at Epcot and walk through.
Valet parking is $25 plus tip, so more than parking at Epcot.
There is no International Gateway theme park parking available, just resort parking for resort guests
The "official answer" is Hollywood Studios. You park there and take the boat to the international gateway. You can also valet at the area resorts but know that during festivals even their valet can be full during the day.