I have a 3yo and 18mo. They have their moments to be certain.
I would have to suggest stopping after dinner sweets. I would give sweets far more sparingly and much earlier in the day. Sugar has a very profound effect on behavior and shouldn't be used a motivator to eat meals.
As far as her defiant behavior: I give my children a count-down to the end of a situation. For example- we do naptime at noon every day. So, 15 minutes before nap I tell the girls (I watch 2 extra girls during the day) "naptime is in 15 minutes. In 15 minutes we're going to cuddle up in our respective beds and rest." Then I remind them at the 10 minute marker, 5 minute marker, and then 2.5 minute marker. I let them know about the process we'll be taking to get ready for naps. For example: "Katie, I'm going to change your diaper in a minute, after I change your diaper we'll have 5 more minutes before sleepy time..." etc for each girl. Then I follow the same protocol each time getting them each to sleep in their beds/rooms.
Some children require this "warning time" to disengage from an activity before they're ready to move on to what needs to be done next.
For your daughter, with leaving the park, let her know when you arrive how long you'll be staying. 15 minutes before it's time to leave start your count-down reminder. At the 2-3 minute marker let her know she has X amount of times on the slide left, or whatever she's playing with. Then let her know at the 1 minute marker you need her to come with you because it's time to leave the park.
One thing I do with my 3yo when we're at the park if she's trying to stay and play- I start walking away. I say "Well, it's time to leave the park Sophia. Mommy and Lily are leaving. I guess we'll see you at home." Then I walk towards my car. She usually comes to us hauling butt.
If she begins to act out and tries to defy you then you need to let her know that you gave her plenty of fair warning. If she continues to defy you let her know that she is making going to the park unpleasant and that you enjoy taking her to the park and would like to continue doing so. Ask if she'd like to return to the park ________ at whatever point in time- when she says yes- say "then it's time to go home. let's go."
When your child is over-tired or hungry all the discipline in the world isn't going to fix the problem. The problem is that the child is hungry/tired. You need to address the source of the problem, not the response to the problem.
If she's tired and not napping- you need to insist on QUIET TIME. Don't call it nap time. Change the name to "Big Girl Quiet time". That's what I do with my 3yo. I tell her "you don't have to sleep, but you do have to lay on your bed and rest your eyes for X amount of time." more often than not she will nap. If she leave her room just walk her back to her bed. Tell her the 1st time - "it's quiet time- you need to lay down and rest your eyes."
If she keeps coming out just pick her up, don't say a word, remain absolutely neutral, and put her back on her bed. Do NOT react. Keep doing it. It may take a few days, but she'll get the point and your lack of reaction will show her you mean business.
As far as hunger goes- offer 3 meals and 2 snacks a day. 16oz max of milk, little to no juice, and as much water as you can get her to drink. Keep her fed on a schedule. Don't cave and don't be a short-order cook. If you have the means to feed her- no child will starve themselves to death by choice. You just need to insist on routine and that you are in charge. By all means give her choices about her food throughout the day, but keep the feeding schedule rather rigid. For example offer her noodles with peas or a nut butter sandwich and grapes for lunch. If she chooses noodles for lunch and refuses to eat them say "tsk tsk tsk, that's too bad sweetie. I guess I'll put this in the sink, and you can have another opportunity to eat at X time."
Do not feel guilty if she refuses food she chooses. She will learn. I choose it, I eat it. Plus, the schedule will allow her body to regulate it's metabolism and all her body to anticipate a certain interval of food. She'll need to learn those rythmns for school when she will be on the schools feeding schedule.
Best wishes!