Can you? Sure, have at it. But, at least right now, parking at DTD is a nightmare. Once the parking garage is open, who knows what the situation will be. Should you? Sure, if that parking fee at the parks is that big a deal to you. But I have to think that if you can't afford to park at WDW, you might want to rethink your vacation plans.
Here's how it will go if you decide to park at DTD and head to a park....arrive at DTD and get a spot. Walk to the bus stop....fair hike..about 10-15 mins. Then, wait for a bus. Depending on time of day that wait could be 10-30+ mins. And the morning buses don't run all that frequently until closer to noon. Then, ride to the resort....10-20 mins. Off the bus, and find transportation to the park. If you are going to MK, then you could take the CR bus, but that will make numerous stops along the way, then you have a 10-15 min walk to the MK. Same thing if you're going to Epcot or DHS...sure, you can walk to either DHS or Epcot. But, the walk to Epcot will take you 10 mins, to DHS will take 15+.
If you decide to take a particular bus, as in the above instances, your wait could be longer. A lot of people just hop on the first resort bus that comes along and then transfer at that resort, usually taking a bus to the park of their choice. So....you can expect to use about 90 mins to get to a park from DTD.
You are better off paying the parking fee and keeping your receipt...you only pay to park once a day. So you can hop to a different park later in the day, show your receipt and you get to just drive in without paying a second time.