"There are no trees in the park" should be accepted.
There's a different nuance here though. Your sentence would be:
공원에 나무 없어 - There are no trees in the park.
Whereas this sentence is - There aren't ANY trees in the park.
They both mean the same thing, but the translations have different nuances.
It is now.
And yet "There are no trees at the park" is not. I will have to report it.
Why does 나무 have 도 after it?
You can find the explanations for the particle 도 here:
"There are not any trees at the park" should be accepted