Your parents should have filed a gift tax return (Form 709) in the year your name was added to the deed. It's likely no taxes would be owed by them, as there's a lifetime exclusion amount in the millions. (You as donee do not need to report the gift and you do not owe taxes it).
If a Form 1099-S was not issued by the settlement agent, then you do not need to report the sale of the home on your income tax return. (You are well under the exclusion amount).
If a Form 1099-S was issued by the settlement agent, then you must report the sale on your income tax return (For 8949 and Schedule D) for the year in which the sale occurred. You will not owe any tax since you are under the exclusion amount, but the sale must be reported since the 1099-S is filed with the IRS and they view that as potentially taxable income (until you tell them otherwise).
Your mother's situation could be the same ... or not. It depends on how long she lived in the house in the years leading up to the sale. In the 5 years preceeding the sale, the house must have been her primary residence for at least 730 days. If that is the case, she qualifies for the $500K exclusion just as you do. If she does not meet the residency requirement, then she doesn't qualify for the exclusion. If she doesn't qualify for the exclusion then she must report the sale regardless of whether a Form 1099-S was issued.
