I think part of the problem that the OP has is a matter of terminology, so I think I'll try and straighten things out.
"Walt Disney World" is the name of the entire 40-something square-mile property in Florida that Disney owns. On this property you find four theme parks (Magic Kingdom, Epcot, Disney's Hollywood Studios and Disney's Animal Kingdom), two water parks (Blizzard Beach and Typhoon Lagoon), the Downtown Disney shopping and entertainment district, and a whole bunch (18+) of hotels. ALLLLL of that is Walt Disney World.
There are quite a few people (especially those who went to WDW when there was only one theme park) who incorrectly refer to the Magic Kingdom as "Walt Disney World". The first park has always been called the Magic Kingdom, and the entire property was WDW. When the property first opened, there was only the MK, two hotels and the campground, so I can see why people might get confused.
Now, on to the tickets. If your father bought 1-day Magic Your Way tickets, they will only be good for getting into *one* of the four theme parks on the day you use it. You need to add the Park Hopper option onto the ticket in order to get into more than one park in one day.
If you want to do one day at the MK and one day at DHS, you'll have to buy a 2-day base ticket.
In terms of actually DOING this, I'd suggest against it unless you REALLY REALLY only want to ride a few key rides in each park. The Magic Kingdom can easily occupy two full days, though you can hit many of the main attractions at least once in a single day. The Studios can also occupy the better part of a full day.
Also be aware that the MK and the Studios are almost 4 miles apart. It takes time to get from one to the other (I'd plan on 45 minutes at the minimum). On a longer vacation where you just want to go to the Studios in the evening to catch Fantasmic and maybe a ride on Tower of Terror, you can "afford" to do that. But if you only have a single day, I'd suggest concentrating on a single park and leave the others for a future visit.
All MYW tickets can be upgraded after you've bought them. You can add more days, the Hopper option, etc. Just stop by a Guest Relations window. You'll pay the difference between what you current have and the price of the new ticket. Just be aware that this has to be done within 14 days from the first *use* of the ticket, or by the end of the day on your ticket day, whichever comes first.
-Rob
