First, you may owe capital gains tax on your gain, but your gain may not be as large as you think.
Your adjusted cost basis is the purchase price plus certain of your closing costs from the purchase that were not deductible, like inspections, transfer taxes, attorney fees, and any bank fees that did not count as points.
You also reduce the selling price by your similar expenses including attorney fees, inspections or transfer taxes paid by the seller, and the real estate commission. All these costs reduce your gain subject to tax.
THEN,
you may qualify for a partial exclusion if you move early due to "unforeseeable circumstances." This includes a job change or change in health that requires a move. One example the IRS gives as qualifying is, you buy a 2 bedroom house and then are pregnant with triplets, and need more room for the family. This rule is covered in IRS publication 523. https://www.irs.gov/forms-pubs/about-publication-523
Your circumstance might qualify as an unforeseeable circumstance depending on your parent's health when you first bought the house. (In other words, if you were already thinking you might have to move mom in soon, then it wasn't unforeseeable. But if your dad's health was good and it looked like they would be able to live independently for a long time, then his death and the change in your mother's circumstance might have been unforeseeable. It depends on the facts of each particular case.)
If you claim the exclusion due to unforeseeable circumstances, you would be able to exlude in your case, up to 18/24ths of the usual limit (in other words, exclude $162K or $375K instead of $250K or $500K.
If you claim the partial exclusion due to unforeseeable circumstances, be sure to keep documentation of that with your other tax papers for at least 7 years.
