OP, you are not accurately stating the rules for claiming your father as a dependent. These rules are on page 21 under step 4 of the 2018 instructions for Form 1040 (
https://www.irs.gov/pub/irs-pdf/i1040gi.pdf). In order:
1. Your father - OK
2. Not qualifying child - Almost certainly not, unless your grandfather's parents are alive - OK
3. Gross income < $4,150 - See below.
4. Provided more than half support - See previous poster's comments and relevant worksheet.
If your father meets all of these criteria, then you can claim him as a dependent relative, which in turn means you can take a $500 dependent tax credit for him and you can also file as HOH.
You also are not accurately stating the rules for the gross income test. Gross income is defined on page 10 of the 2018 instructions for Form 1040. Quoting in relevant part:
"*Gross income means all income you received in the form of money, goods, property, and services that isn't exempt from
tax, including any income from sources outside the United States or from the sale of your main home (even if you can
exclude part or all of it). Don’t include any social security benefits unless (a) you are married filing a separate return and
you lived with your spouse at any time in 2018 or (b) one-half of your social security benefits plus your other gross
income and any tax-exempt interest is more than $25,000 ($32,000 if married filing jointly)."
Assume your parents file MFJ. Then you would not include his SS benefits because the last sentence says to exclude them unless (a) or (b) applies. (a) does not apply because they are MFJ not MFS, and (b) does not apply because SS plus their other income is only $31K, and $31K is less than the $32K limit for MFJ.
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Note that if you claim your father as a dependent, I believe that will affect how he files, because he will have to check the box at the top of Form 1040 saying that someone else can claim him as a dependent. This in turn will affect the standard deduction that he can claim on line 8. He'll have to use the worksheet on page 35 of the instructions.
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As a further aside, a couple can choose to file MFJ or MFS regardless of their living situation. You can file MFJ even if you don't live together, and you can live together the whole year and file MFS. The general strategy is to evaluate filing both ways and file the way that gives you the best result. Most often MFJ beats MFS, but there are limited situations where MFS wins. See page 15 of the 2018 instructions for Form 1040 under Filing Status.