U.S. Citizen living in Europe wanting to register a consulting company in Oklahoma
Your plan would work as long as the laws in the European country where you are based at do not require you to register your US company in the local jurisdiction.
You could register an LLC in Oklahoma, and use your dad as the registered agent, provided he is available at the registered address during business hours. You can also open a bank account for that LLC, provided you can travel to the US to do that, or have your dad open one instead as a company representative. You would then be able to ask your clients to pay your company by wiring funds to your US bank account.
Again, I can't really advise you on registration requirements in your European country, however since you are an American you are required to pay taxes in the US anyway, so you might as well do business using a US registered LLC. I would also recommend you to consult with a local accountant to see if there are any caveats we didn't think or know of.