I don't think this is unethical. In fact, it sounds like you are doing competent work and were doing this before your father became director.You have done nothing wrong and your father is not doing anything wrong by keeping a competent designer, who happens to be his son, on board with his team. This is just how the world works sometimes. There is nothing wrong with your dad keeping a door open for you (imagine being in his shoes) and there is nothing wrong with you seizing a career opportunity. I would suggest talking to your dad about it and being open and honest with him about your feelings. Clear communication is the best tool in almost every scenario in life and one of the things that most humans do very badly.

This will take time and maturity to realize (I am assuming you are a younger designer based on your worries), but try not to let the feelings of others (the grumblers) dictate the direction of your career. In fact, you said that this is only a worry of yours. If you suspect others are grumbling about your opportunity, don't let it get you down, especially if there is no proof that they are grumbling! If it's not happening, then don't worry about it! That is an irrational fear that you have created in your mind and you are psyching yourself out because (maybe?) you might not feel worthy. You sound like a typical designer! :D Keep your chin up and keep doing competent work. Prove that you are worth it by helping the company succeed. Document your successes along the way and prove your worth when review time comes around. If the family/business relationship starts to bother either of you, then consider other job options. Give your father that out as well. Just be open and honest and keep business business.

In time, you will learn that people love to complain about anything they can complain about. It's your duty as a designer to rise above this negativity and always do your best work. That is one of the biggest challenges we face every day. If your dad is the director, he makes the decisions. The feelings of others about your "worth" do not matter because ultimately, he makes the decisions and it is their job to work with his direction. They always have the option to leave if they do not agree with corporate decisions. Also, cheaper designers are not necessarily a good thing and it's probably a very good thing that he sees the worth of your work enough to put his reputation on the line for you.

At the end of the day, you are constantly building relationships. People will remember you for your attitude, your contribution to your company, and the quality of your work. Not your cost. Good luck and congratulations!