A. Assume your parent doesn't have an estate worth $5 million or higher, you do not have to worry about inheritance tax.
For capital gain tax, when your parent dies and you inherit the house, the step-up value of the house ($500,000) is your cost. If you sell the house right away without any further gain, you do not have any capital gain tax. If you keep the house and sell it in the future, any gain above $500,000 and your further investment in the house will lead to capital gain tax.