Taking her to the park becomes available after you have gone on a couple of dates with her, walked her back to her dorm, and kissed her.
Make sure you're calling her in the evening for the date. I think you can only set up coffee dates when you talk to her in the morning in the lobby, but if you call her in the evening, you should be able to take her to the movie theater as well.
Check the hint screen and see if it helps.
If you're still stuck, getting on the discord is the best way to get help. You can post your save file there and we can see if you've found a bug or anything.