That is true, the park is generally just open on the weekends. What email address did you write to?
The full list of contacts is here: http://www.provincia.fi.it/pratolino...ici-e-servizi/
but I don't see anything for "visits".
I would try this one but to be honest, not sure how willing they are to schedule openings for a few people. I know that they do so for groups, not sure what the minimum is though.
Rita Poli - Telefono: 055-2761419 - mail: email@example.com