I am glad you have a boss/friend who is supportive, and proactive.
I don't know your situation but you need a break from mom & dad obligations for sure, or to be able to have control over that part of your life.
Your parents might think they are doing the right thing (keeping an eye on you, keeping you "busy" who knows....but this does seem to be more for them than for you as they have not (?) asked you what would help.
As for dad's driving. You could be up front. Or, you could have another place to go "after" or "before" that requires you to travel solo & meet them. It is ok, also to say that you are sorry but cannot make lunch.