"1. Since I am filing for 2020 when they did not have Medicaid, it should not affect the fact they got Medicaid approval in 2021, right? "
Plus review the requirements to claim an older relative as a dependent (a "qualifying relative") to be Head of Household on Table 2-1 on page 25 in IRS Pub 17 and Table 3-1 on page 26 (to see who is a qualifying relative). Specifically, they cannot make more than $4,300 (each) and you must provide half of their support.
If you satisfy these tests, then you may be able to file as Head of Household because you have two qualifying dependents.
"2. Will it affect Medicaid eligibility for 2021? Shall I NOT file them as dependent in 2021? " We address income tax issues on this forum. Medicaid not only has its own rules but they vary from state to state.
I would encourage to contact someone who deals with Medicaid in Illinois for the answer to question #2.
@drhellga
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