I find public transportation is also subsidized.

The subsidies for cars is quite high, and it ought to be low because it is a destructive form of transportation. The subsidies for transit is quite low and should be high because it is the superior solution to the problem of moving people around.

Any attempt to increase fares to cover costs gets a huge amount of push-back.

Car roads having fares to cover costs isn't even a part of the discussion in the US. User fees (mostly gas tax) account for ~1/3 of the cost of roads, and this percentage is declining ^source^. That means people who make the ethical choice of not using a car are paying for those who make the unethical choice of using a car.

In general. I think it is good for the tax code to encourage prosocial behavior. Right now it does the opposite.

People already pay for parking except on private lots

This does not match with my experience. Where I live, and almost everywhere I have been, curb parking is usually free. And when its not free, it is highly discounted from the price of the land if you were to use the land for any other purpose.

Also, there are a lot of private lots. This is usually due to the strict parking mandates, where the government forces developers to build parking lots. This leads to a parking abundance where drivers refuse to pay reasonable fees for parking.

I recommend Henry Grabar's recent book, Paved Paradise on the topic of parking.