lesschumras, whether right or wrong, the parents fear isn’t to the extent of “what they’ll do”, it is who they might choose to associate with, that might displease the parents and “ruin” reputations and chances of a good Shidduch, by their daughters simply speaking to the wrong people.
Although, with so many years of the Shidduch crisis having left us with many many disgruntled older single girls around putting a damper on the home atmosphere, I think now many parents of girls almost wish their daughters could meet boys on their own.