I really like Jen's suggestion. When it's time to quit swinging take the time to get her involved in something else that's tactile. Sand is good but the park may not have a sand box. My grandkids like to pick up the wood chips and toss them in the air. This is OK when there aren't a lot of kids around. Sliding down the slide might help. She could probably learn how to do that on her own after you show her how several times. I've noticed that most kids, even those that young, like to run. They chase each other. Perhaps you could have her chase another child or chase her yourself.
Perhaps frequently talking with her during the times you're not at the park might help. Explain why it's important to share, that you get tired pushing her, etc. and ask her what she'd like to do when she get's off the swing. Find a book about sharing swing time and/or make up a story with her. Make her the main character, let her say the words about wanting to swing and how she feels when she has to stop. Stories often help children work thru situations.
What do you do when she won't stop crying, screaming? Perhaps you can do that before she gets going full force. Most of all, tho, never put her back on the swing. This teaches her that if she makes a big enough fuss she gets what she wants. Divert her attention by getting her started on something else or by holding her and being sympathetic to her feelings.
It may also be that, after you've tried various things, that you'll have to let her cry. It's OK to be disappointed and to express that disappointment. Soothe her with your voice while you're doing something else.
It will work best if you can stop being anxious about what other people think. If she is throwing a temper tantrum, other mothers know that to not pay attention is the way to go. Just because you do daycare does not mean that you know magic! Do what is the best thing for you and your daughter. I suspect that there will be more people sympathetic to your situation and acknowledging that you have to ignore her than those who will be critical.