Is it unhealthy not to wear shoes?!

Indoor allergens, as far as I know, are usually problematic when inhaled. I'm assuming that you are referring to walking barefooted. So, I would expect that you would get a local reaction on your feet if you were walking on something to which you were allergic. If not then walking barefoot in the house is probably OK in terms of your allergies.
If you know the specific agents to which you are allergic then you could research it or ask your family doctor.

Walking barefoot outside is a different matter if you are prone to allergic reactions!

If your feet are not uncomfortably cold then it is probably OK. If you do feel generally cold then this isn't good for your health. You can lose quite a bit of heat through the surface area of your feet. In this case, perhaps wear some lightweight socks. You will hold in some warmth and still retain the feel of not wearing shoes or slippers when you walk around.

Walking barefoot can be very beneficial to the overall health of you feet in many cases as well as feeling comfortable and relaxing. Some underlying conditions make it inadvisable though so your Dad is right to be concerned.

The other side to all this is his house - his rules. Sorry but I had to say it.
Why not do some research of your own and then present it honestly to your Father, I'm sure he has your best interests at heart.

Best wishes.

People mainly wear slippers around the house so that their feet won't get cold. I'm not sure about it helping out with your allergies, but it might. If you're feet are warm, I think it might slow them down a bit, but then again that's just an educated guess, I don't have allergies, so I wouldn't know.

I am not agree with your father