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The estate doesn't need to issue the form 1099-misc to you are the executor of your parents estate.
For reporting your income - please see for reference IRS publication 525 page 33 - http://www.irs.gov/pub/irs-pdf/p525.pdf
Personal representatives. All personal representatives must include in their gross income fees paid to them from an estate. If you are not in the trade or business of being an executor (for instance, you are the executor of a friend's or relative's estate), report these fees on Form 1040, line 21. If you are in the trade or business of being an executor, report these fees as self-employment income on Schedule C or Schedule C-EZ (Form 1040). The fee is not includible in income if it is waived.
For state income tax purposes - you will claim this income for the state where you are a resident.
You may pay estimate taxes at any time, but you still might owe underpayment penalties depending on your last year tax liability. To determine if there will be any penalty - please fill the form 2210 - http://www.irs.gov/pub/irs-pdf/f2210.pdf
If there is no underpayment penalty - you may pay all tax liability by Apr 15.