Salaam,
No, a girl cannot get married without her parent's permission. She can if she wanted to, but it is not allowed. If a girl wants to get married, first she has to have her father's approval that she can get married to the guy. Secondly there has to be her father to help pay foe the wedding. Thirdly, her father has to be there in order to be her Wali. Without any of these reasons, then how can she get married without her father? In addtion to that if you think about it what if she gets divorced and then her husband send her back to her fathers's house. Without her father knowing she may never even be allowed to come back to her father's house.
Wasalaam.
I think permission and involvement of walis (guardians) is a religious necessity in case of both the man and the woman. But if two youths marry themselves secretly beyond the knowledge of their respective walis, they will be sinners for violation of the religious manners. But will the marriage be valid? In that case, what can the walis do to their recalcitrant offsprings?