How old are you and are you paying the operating costs of the flights?
Reason: If you’re a minor, your parents paying the bill is definitely not the sort of violation the FAA is looking for. If not a minor, you may want to set up an arrangement where dad pays you for whatever he pays you for (not flying) and you pay the operating costs, to be perfectly legal.
As long as nobody is paying you to fly anywhere as a Private pilot certificate holder, you’re fine. (There are exceptions to this, also. They probably don’t apply to your description of your flights.)
Your flight instructor will go over all of these regulations and you’ll be tested on them both in written form and verbally at your practical test for your certificate long before you’ll need to worry about it much.
You said you weren’t concerned about insurance but your dad and the instructor will have to have a talk about insurance for the instructor if you don’t already hold a Private certificate and require training. The instructor will have significant liability teaching you in a privately owned aircraft.