Q. My question is can our ownership qualify as inheritance or is it gift?
A. It's a gift.
The cost basis of a gift is the giver's cost basis.
The good news is that your father's cost basis is the fair market value on your mother's date of death in 2020.
The usual way it works is that only your mother's half of the cost basis "steps up". But, because CA is a community property state, the entire cost basis steps up. So, your capital gain is 25% of difference between the sale price and the value in 2020 (not what your parents paid for it in 2013). If that results in a loss (considering the expenses of sale), you are not allowed to claim the capital loss as a deduction, because it is personal use property (a relative lives in it).
Re your comment, "My sister and I just received ownership this year so we do not quality for $250K Home Sale Tax exclusion": you do not qualify for the home sale exclusion, regardless of timing, because it was not YOUR primary residence.
