Im the same way.
My daughter is a little shy anyway so I make sure she knows I am confident in her abilities so that she feels confident as well.
And they do pick up on that. My OH is a little (ok quite a lot) helicopter-y and I do tell him to relax and trust her to try things but its something he struggles with. But I do feel like when were at the park together she picks up on his fear more than my confidence and as a result doesnt do things alone that I know she can and has already done.
OH is already trying to work on it and I keep telling him that he had bruises and scrapes as a child and that didnt mean his mom was a bad mom at all. And he gets it, in theory. But still as soon as DD is up and playing on the playground especially with older kids, he hovers around her like.. well a helicopter