Death means the end of many things, but it won't stop is the Internal Revenue Service from expecting any income taxes the decedent owed to be paid from the decedent's estate. According to the IRS, if a return is required, its the responsibility of the executor of the decedent's estate. If you are the executor of your parent's estate, the IRS will expect a return from you.
As the executor, you must determine whether a final income tax return is required for your deceased parent. A return is required if your parent would have been required to file if they had lived until the end of the year. For example, for the 2014 tax year, if your parents was single and over 65, you must file a final income tax return if your parent earned $11,500 in the calendar year in which your parent died.