You do not report your dependent's income on your tax return. You also do not get to take credit for your dependent's refund (or withholding or estimated payments) on your tax return. Your mother gets full credit for that on her return, since your father (apparently) will not be filing a return.
If your father qualifies as your dependent (and its not clear that he does), you may deduct most of his nursing home costs as medical expenses. Medical expenses are only deductible as an itemized deduction, subject to a 7.5% of AGI threshold.
You should not be claiming your father as a dependent until you and you mother have prepared comparative returns to see which way the family comes out best. By forcing your mother to file as Married Filing Separately (MFS) (instead of Married Filing Jointly (MFJ) ), she will almost assuredly lose more than you gain. You can use this tool: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1
The brokerage account further complicates things, even though it's in her name an SS#, your father may be responsible for reporting half the income.
