My first was also extremely cautious and I was guilty of not watching her one day and she fell badly off equipment and was bleeding. I saw it from a distance and ran over but felt awful. One of the mums who was helping her asked me if I was the nanny and I thought it was because I looked so young 😎 but then I realised it was because I was being so inattentive.
So I watched more closely after that. But then I had no 2, who was a daredevil but extremely capable so little by little I let go. No 3 is more like his sister and wanted me close but is more independent now and no 4 couldn't care less if I was even at the park let alone beside her. She smacks my hand away when I try and help her and tells me to "go!" When I try and stand close.
Every child is different and I'd trust the parents to know their own kids and their capabilities.