We are staying off site for the first night then checking to a Disney resort in the morning. If we arrive at a decent hour, we are considering going to DTD to look around. Are there parking fees?
In the morning you'll never have a problem finding a spot. It becomes harder as the day progresses. There is seldome a problem finding a spot over on the west side. I go to House of Blues at least a couple times a month and can always find parking in front of Cirq.
All the lots are fairly empty around 9-10. After 12 they start to fill up more. Depending on the day, it may take a while to find a spot after 3 near the marketplace but Pleasure Island and West Side parking is generally available.