Date - yes.
Do anything even remotely sexual - no,
As a poster on another board frequently says, if you couldn't do it in the front pew of the church on Christmas morning with your parents sitting in the pew behind you, it's off limits.
The above assumes that your parents really have given you permission. If at any time they say no, then the answer automatically becomes no. They cannot make the illegal, legal but they can make the legal illegal. Permission can be withdrawn at any time until YOU turn 18. Until that time, you may see and have contact ONLY with those they have given you permssion to see and have contact with.