Do we still park in the old Downtown Disney parking, over near the Disneyland Hotel; or is there a new lot to park in due to the construction?

The downtown Disney parking lot is still there

Just go with signs there, it's very obvious

In the old one

That is the most convenient, but depending on what day u go u may want to park & use the shuttle.

Same parking structure like Rebecca said. The construction is happening by the Mickey and Friends parking structure