As others have said, you can use Starbucks gift cards to pay for purchases in the parks. You can also use the Starbucks App and earn Rewards (if you've joined the Rewards program). However, you cannot redeem rewards at the in-park locations.
Yes the in-park locations take the giftcards. If you are a member of the rewards you will have to use the DtD location to redeem them. The in parks locations also do not participate in any discounts & promotions (such as the recent BOGO on holiday drinks), again you will have to visit the DtD location for that.
You also only get stars for in park purchases. If there's a promotion like buy 3 frappuccinos and get 50 bonus stars you won't get credit for what items you purchased, just stars for the total amount of the purchase.
DTD, and for the one in the middle of DTD, if you do that, go in the *side* door. It's through the outdoor seating area. If you try to go in the main door you'll get stuck on a busybusy day and won't be able to move to the back to where the pickup area is.