Let's sort out the grammar issue that's the same in both these sentences, first of all. By 'park', do mean an outdoor leisure area, with trees and so on? If so, both of these sentences need an article. Have you studied about how to use articles? Remember that 'park' is a countable singular noun, so you can never say 'to park'. You have to say, 'to a park' or 'to the park' or 'to my local park', for example.
Now for the tenses:
1. "I went to the park today" is fine. You might say this when you are at home in the evening. A family member says "Where did you go today?" and you could answer "I went to the park today". This is an example of the past simple tense. No problem there.
2. "I go to the park today" would be a strange thing to say. It is grammatically acceptable, but very unusual. [The only time you could say this is if you are reading from strict timetable where you have to go a different location every day, and the event scheduled for today will be a visit to the park. This seems unlikely].
It's hard to see what you mean by "I go to the park today". If a native English speaker heard or saw this statement, we would not fully understand it. This is because the time is not clear.
What idea do you have in mind when you write "I go to the park today" ? When are you saying this?
Is it BEFORE your trip to the park? If so, you would need to use a future tense: "I'm going to the park today" or "I'm going to go to the park today". Or if you've just made this decision, you could also say "I'll go to the park today".
Or are you on your way to the park? If you meet a friend while you are walking to the park, you could say "I'm going to the park".
Finally, if you say this AFTER your trip to the park, you'll need to use a tense which shows that this activity was in the past. You either use a present perfect "I've been to the park today" or the past simple, as in your sentence above.
I hope that all makes sense.